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{{distinguish|Reanalysis}}
{{short description|Mathematics of real numbers and real functions}}
{{short description|Mathematics of real numbers and real functions}}
{{More citations needed|date=October 2024}}
{{distinguish|Reanalysis (disambiguation){{!}}Reanalysis}}
{{Math topics TOC}}
In [[mathematics]], the branch of '''real analysis''' studies the behavior of [[real number]]s, [[sequence]]s and [[Series (mathematics)|series]] of real numbers, and [[real function]]s.<ref>{{Cite web|url=https://www.math.ucla.edu/~tao/resource/general/131ah.1.03w/week1.pdf|title=Lecture notes for MATH 131AH|last=Tao|first=Terence|date=2003|website=Course Website for MATH 131AH, Department of Mathematics, UCLA}}</ref> Some particular properties of real-valued sequences and functions that real analysis studies include [[Convergent series|convergence]], [[limit (mathematics)|limits]], [[continuous function|continuity]], [[smoothness]], [[derivative|differentiability]] and [[integral|integrability]].


Real analysis is distinguished from [[complex analysis]], which deals with the study of [[complex number]]s and their functions.
[[File:Illustration of supremum.svg|thumb|300px|The [[least-upper-bound property]] (completeness) of the real numbers: that every non-empty bounded above subset has a [[infimum and supremum|supremum]]. Virtually every major theorem in real analysis relies on the [[completeness of the real numbers]], including those concerning the existence of limits of sequences, the behaviour of continuous functions, and the existence and uniqueness of solutions to differential equations.]]
 
'''Real analysis''' is the part of [[mathematical analysis]], especially as taught in undergraduate and graduate courses, that develops [[calculus]] rigorously over the [[real number]]s and [[Euclidean space]]s. Introductory real analysis is sometimes called '''advanced calculus''', and studies [[limit (mathematics)|limits]], [[continuous function|continuity]], [[compact space|compactness]], [[derivative|differentiation]], [[integral|integration]], and [[series (mathematics)|series]]. More advanced courses often include [[measure theory]], [[Lebesgue integration]], and [[function space]]s. Real analysis is also known, especially in older books, as the theory of '''functions of a real variable''', in contrast to the theory of [[complex variable]]s.


==Scope==
==Scope==
{{cleanup|section|reason=This section goes too heavily into detail about each concept. It should just portray a brief overview in relation to the field of real analysis|date=June 2019}}
===Real numbers===
 
The [[real number]]s are the basic setting of real analysis, which begins with [[construction of the real numbers|their construction]]. The real numbers are distinguished from the [[rational number]]s by their ''completeness''. Roughly speaking, the real numbers have no gaps. This completeness can be formalized in several equivalent ways, one of which is the [[least upper bound property]]. This states that if a non-empty set of real numbers is [[bounded set|bounded]] above, meaning that all of its elements are less than some number (an upper bound), then there is a ''least'' upper bound, that is an upper bound that is smaller than all of the others.{{sfn|Rudin|1976|loc=Ch. 1}}
===Construction of the real numbers===
{{Main|Construction of the real numbers}}
The theorems of real analysis rely on the properties of the (established) [[real number]] system. The real number system consists of an [[uncountable set]] (<math>\mathbb{R}</math>), together with two [[binary operation]]s denoted {{math|+}} and {{math|-}}, and a [[total order]] denoted {{math|≤}}. The operations make the real numbers a [[Field (mathematics)|field]], and, along with the order, an [[ordered field]]. The real number system is the unique ''[[Completeness (order theory)|complete]] ordered field'', in the sense that any other complete ordered field is [[Isomorphism|isomorphic]] to it. Intuitively, completeness means that there are no 'gaps' (or 'holes') in the real numbers. This property distinguishes the real numbers from other ordered fields (e.g., the rational numbers <math>\mathbb{Q}</math>) and is critical to the proof of several key properties of functions of the real numbers. The completeness of the reals is often conveniently expressed as the ''least upper bound property'' (see below).
 
===Order properties of the real numbers===
The real numbers have various [[lattice theory|lattice-theoretic]] properties that are absent in the complex numbers. Also, the real numbers form an [[ordered field]], in which sums and products of positive numbers are also positive. Moreover, the ordering of the real numbers is [[totally ordered|total]], and the real numbers have the [[least upper bound property]]: <blockquote>''Every nonempty subset of <math>\mathbb{R}</math> that has an upper bound has a [[Supremum|least upper bound]] that is also a real number.'' </blockquote>These [[partially ordered set|order-theoretic]] properties lead to a number of fundamental results in real analysis, such as the [[monotone convergence theorem]], the [[intermediate value theorem]] and the [[mean value theorem]].


However, while the results in real analysis are stated for real numbers, many of these results can be generalized to other mathematical objects. In particular, many ideas in [[functional analysis]] and [[operator theory]] generalize properties of the real numbers – such generalizations include the theories of [[Riesz space]]s and [[positive operator]]s. Also, mathematicians consider [[real part|real]] and [[imaginary part]]s of complex sequences, or by [[strong operator topology|pointwise evaluation]] of [[operator (mathematics)|operator]] sequences.{{Clarify|date=June 2021}}
Most of the theorems that are proved in real analysis rely on completeness in one way or another. Some examples where its relevance are most apparent are as follows. The convergence of bounded [[monotone sequence]]s, that is sequences that are increasing (or decreasing), is essentially equivalent to the least upper bound property, stated in sequence form. Completeness is also reflected in the [[intermediate value theorem]]: the continuous image of an interval is again an interval, so continuous functions cannot create gaps.{{sfn|Abbott|2015|loc=Chs. 1, 4}}{{sfn|Pugh|2015|loc=Chs. 1–2}}
 
=== Topological properties of the real numbers ===
Many of the theorems of real analysis are consequences of the topological properties of the real number line. The order properties of the real numbers described above are closely related to these topological properties. As a [[topological space]], the real numbers has a ''standard topology'', which is the [[order topology]] induced by order <math><</math>. Alternatively, by defining the ''metric'' or ''distance function'' <math>d:\mathbb{R}\times\mathbb{R}\to\mathbb{R}_{\geq 0}</math> using the [[absolute value]] function as {{nowrap|<math>d(x, y) = |x - y|</math>,}} the real numbers become the prototypical example of a [[metric space]]. The topology induced by metric <math>d</math> turns out to be identical to the standard topology induced by order <math><</math>. Theorems like the [[intermediate value theorem]] that are essentially topological in nature can often be proved in the more general setting of metric or topological spaces rather than in <math>\mathbb{R}</math> only. Often, such proofs tend to be shorter or simpler compared to classical proofs that apply direct methods.
 
===Sequences===
{{Main|Sequence}}
A '''''sequence''''' is a [[function (mathematics)|function]] whose [[domain of a function|domain]] is a [[countable]], [[totally ordered]] set.<ref name="Sequences intro">{{cite web |title=Sequences intro |url=https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:sequences/x2f8bb11595b61c86:introduction-to-arithmetic-sequences/v/explicit-and-recursive-definitions-of-sequences |website=khanacademy.org}}</ref> The domain is usually taken to be the [[natural number]]s,<ref name="Gaughan">{{cite book|title=Introduction to Analysis |last=Gaughan |first=Edward |publisher=AMS (2009)|isbn=978-0-8218-4787-9|chapter=1.1 Sequences and Convergence|year=2009 }}</ref> although it is occasionally convenient to also consider bidirectional sequences indexed by the set of all integers, including negative indices.
 
Of interest in real analysis, a '''''real-valued sequence''''', here indexed by the natural numbers, is a map <math>a : \N \to \R : n \mapsto a_n</math>.  Each <math>a(n) = a_n</math> is referred to as a '''''term''''' (or, less commonly, an '''''element''''') of the sequence.  A sequence is rarely denoted explicitly as a function; instead, by convention, it is almost always notated as if it were an ordered ∞-tuple, with individual terms or a general term enclosed in parentheses:<ref>Some authors (e.g., Rudin 1976) use braces instead and write <math>\{a_n\}</math>.  However, this notation conflicts with the usual notation for a [[Set theory|set]], which, in contrast to a sequence, disregards the order and the multiplicity of its elements.</ref>
<math display="block">(a_n) = (a_n)_{n \in \N}=(a_1, a_2, a_3, \dots) .</math>
A sequence that tends to a [[Limit (mathematics)|limit]] (i.e., <math display="inline">\lim_{n \to \infty} a_n</math> exists) is said to be '''convergent'''; otherwise it is '''divergent'''.  (''See the section on limits and convergence for details.'')  A real-valued sequence <math>(a_n)</math> is '''''bounded''''' if there exists <math>M\in\R</math> such that <math>|a_n|<M</math> for all <math>n\in\mathbb{N}</math>.  A real-valued sequence <math>(a_n)</math> is '''''monotonically increasing''''' or '''''decreasing''''' if
<math display="block">a_1 \leq a_2 \leq a_3 \leq \cdots</math>
or
<math display="block">a_1 \geq a_2 \geq a_3 \geq \cdots</math>
holds, respectively.  If either holds, the sequence is said to be '''''monotonic'''''.  The monotonicity is '''''strict''''' if the chained inequalities still hold with <math>\leq</math> or <math>\geq</math> replaced by < or >.
 
Given a sequence <math>(a_n)</math>, another sequence <math>(b_k)</math> is a '''''subsequence''''' of <math>(a_n)</math> if <math>b_k=a_{n_k}</math> for all positive integers <math>k</math> and <math>(n_k)</math> is a strictly increasing sequence of natural numbers.


===Limits and convergence===
===Limits and convergence===
{{Main|Limit (mathematics)}}
The concept of a [[limit (mathematics)|limit]] underlies many of the ideas of [[calculus]], such as the [[derivative]]. It is therefore of fundamental importance in real analysis, which provides the rigorous justification for calculus.  Limits describe how a sequence, function, or family of functions behave under a limiting process, such as letting an index tend to infinity, or a letting point become very large or approach another point. The formal language of limits defines continuity, differentiation, integration, infinite series, and various kinds of approximations and asymptotics.{{sfn|Abbott|2015|loc=Chs. 2, 4–7}}{{sfn|Pugh|2015|loc=Chs. 2–5}}{{sfn|Rudin|1976|loc=Chs. 3–7}}
Roughly speaking, a '''limit''' is the value that a [[function (mathematics)|function]] or a [[sequence]] "approaches" as the input or index approaches some value.<ref>{{cite book|last=Stewart|first=James|author-link=James Stewart (mathematician)|title=Calculus: Early Transcendentals|publisher=[[Brooks/Cole]]|edition=6th|year=2008|isbn=978-0-495-01166-8|url=https://archive.org/details/calculusearlytra00stew_1}}</ref> (This value can include the symbols <math>\pm\infty</math> when addressing the behavior of a function or sequence as the variable increases or decreases without bound.)  The idea of a limit is fundamental to [[calculus]] (and [[mathematical analysis]] in general) and its formal definition is used in turn to define notions like [[continuous function|continuity]], [[derivative]]s, and [[integral]]s. (In fact, the study of limiting behavior has been used as a characteristic that distinguishes calculus and mathematical analysis from other branches of mathematics.)


The concept of limit was informally introduced for functions by [[Isaac Newton|Newton]] and [[Gottfried Wilhelm Leibniz|Leibniz]], at the end of the 17th century, for building [[infinitesimal calculus]]. For sequences, the concept was introduced by [[Augustin-Louis Cauchy|Cauchy]], and made rigorous, at the end of the 19th century by [[Bernard Bolzano|Bolzano]] and [[Karl Weierstrass|Weierstrass]], who gave the modern [[ε-δ definition]], which follows.
A recurring problem in real analysis is not just whether a limit exists, but how well one object approximates another. A convergent sequence approximates its limit, or a differentiable function is approximated by a linear function coming from the derivative. But real analysis can provide not only tools to justify the existence of these limits and how to calculate them, but also quantitative estimates of how good the approximation is. One example of this is the [[Taylor's theorem|Taylor remainder]], which gives an effective and computable constant that determines how well the linear approximation (or higher-order [[Taylor polynomial]]) approximates a function on an interval.{{sfn|Abbott|2015|loc=Ch. 5}}{{sfn|Pugh|2015|loc=Ch. 3}}{{sfn|Rudin|1976|loc=Ch. 5}}


'''Definition.''' Let <math>f</math> be a real-valued function defined on {{nowrap|<math>E\subset\mathbb{R}</math>.}} We say that '''''<math>f(x)</math> tends to <math>L</math> as <math>x</math> approaches <math>x_0</math>''''', or that '''''the limit of <math>f(x)</math> as <math>x</math> approaches <math>x_0</math> is <math>L</math>''''' if, for any <math>\varepsilon>0</math>, there exists <math>\delta>0</math> such that for all <math>x\in E</math>, <math>0 < |x - x_0| < \delta</math> implies that <math>|f(x) - L| < \varepsilon</math>. We write this symbolically as
For sequences of functions, real analysis distinguishes between different modes of convergence.  A sequence of functions [[pointwise convergence|converges pointwise]] if it converges at every point, but, roughly speaking, the rate of convergence may vary from point to point. It [[uniform convergence|converges uniformly]] if it converges at all points at a comparable rate. Uniform convergence can be understood intuitively in terms of the graphs of the functions in the sequence: it means that, for any given thin error band around the limiting function, all but finitely many functions in the sequence stay within the band. Pointwise convergence means that this is true for an error band around each point, but the finite set of functions which must be excluded varies from point to point.{{sfn|Abbott|2015|loc=Ch. 6}}{{sfn|Rudin|1976|loc=Ch. 7}}
<math display="block">f(x)\to L\ \ \text{as}\ \ x\to x_0 ,</math> or as <math display="block">\lim_{x\to x_0} f(x) = L .</math>
Intuitively, this definition can be thought of in the following way: We say that <math>f(x)\to L</math> as <math>x\to x_0</math>, when, given any positive number <math>\varepsilon</math>, no matter how small, we can always find a <math>\delta</math>, such that we can guarantee that <math>f(x)</math> and <math>L</math> are less than <math>\varepsilon</math> apart, as long as <math>x</math> (in the domain of <math>f</math>) is a real number that is less than <math>\delta</math> away from <math>x_0</math> but distinct from <math>x_0</math>.  The purpose of the last stipulation, which corresponds to the condition <math>0<|x-x_0|</math> in the definition, is to ensure that <math display="inline">\lim_{x \to x_0} f(x)=L</math> does not imply anything about the value of <math>f(x_0)</math> itself. Actually, <math>x_0</math> does not even need to be in the domain of <math>f</math> in order for <math display="inline">\lim_{x \to x_0} f(x)</math> to exist.


In a slightly different but related context, the concept of a limit applies to the behavior of a sequence <math>(a_n)</math> when <math>n</math> becomes large.
For sequences of functions, pointwise convergence often fails to preserve operations on the limit function. For example, it is not generally true that the pointwise limit of a sequence of continuous functions is continuous, or that the integral of the functions in a sequence passes to the integral of the limit function.  But the ''uniform'' limit of continuous functions is continuous, and one can exchange integration and uniform limits on suitable domains.  Uniform convergence is therefore important for many applications of real analysis. Questions such as "when is [[differentiation under the integral]] allowed?" or "when can I integrate an infinite sum term-by-term?" are typical examples in which uniform convergence provides a simple answer.{{sfn|Abbott|2015|loc=Ch. 6, 8}}{{sfn|Rudin|1976|loc=Ch. 7}}


'''Definition.''' Let <math>(a_n)</math> be a real-valued sequence. We say that <math>(a_n)</math> '''''converges to''''' <math>a</math> if, for any <math>\varepsilon > 0</math>, there exists a natural number <math>N</math> such that <math>n\geq N</math> implies that <math>|a-a_n| < \varepsilon</math>. We write this symbolically as
===Differentiation and regularity===
<math display="block">a_n \to a\ \ \text{as}\ \ n \to \infty ,</math>or as<math display="block">\lim_{n \to \infty} a_n = a ;</math>
[[Derivative|Differentiation]] measures the local rate of change of a function.  In one variable, the derivative gives the slope of the best linear approximation to a function near a point. This point of view extends to several variables, where differentiability is expressed in terms of approximation by a linear map.{{sfn|Abbott|2015|loc=Ch. 5}}{{sfn|Pugh|2015|loc=Ch. 3}}{{sfn|Rudin|1976|loc=Chs. 5, 9}}
if <math>(a_n)</math> fails to converge, we say that <math>(a_n)</math> '''''diverges'''''.


Generalizing to a real-valued function of a real variable, a slight modification of this definition (replacement of sequence <math>(a_n)</math> and term <math>a_n</math> by function <math>f</math> and value <math>f(x)</math> and natural numbers <math>N</math> and <math>n</math> by real numbers <math>M</math> and <math>x</math>, respectively) yields the definition of the '''''limit of <math>f(x)</math> as <math>x</math> increases without bound''''', notated <math display="inline">\lim_{x \to \infty} f(x)</math>. Reversing the inequality <math>x\geq M</math> to <math>x \leq M</math> gives the corresponding definition of the limit of <math>f(x)</math> as <math>x</math> ''decreases'' ''without bound'', {{nowrap|<math display="inline">\lim_{x \to -\infty} f(x)</math>.}}
In addition to proving rigorously the fundamental properties of the derivative from calculus, such as the [[chain rule]], real analysis also establishes the main theorems about the derivative, such as the [[mean value theorem]] and some of its generalizations like the [[Cauchy mean value theorem]]. Roughly speaking, the mean value theorem relates the derivative of a function to its average rate of change over intervals. The mean value theorem is important because it leads to an effective way to estimate the error in certain approximations or averaging processes.{{sfn|Abbott|2015|loc=Ch. 5}}{{sfn|Pugh|2015|loc=Ch. 3}}{{sfn|Rudin|1976|loc=Ch. 5}}


Sometimes, it is useful to conclude that a sequence converges, even though the value to which it converges is unknown or irrelevant. In these cases, the concept of a [[Cauchy sequence]] is useful.
Real analysis studies not only existence of derivatives, but also degrees of regularity.  A function may be continuous but nowhere differentiable, differentiable but not continuously differentiable, or smooth (having derivatives of all orders) but not analytic (equal to its Taylor series). One theorem that is sometimes shown at the elementary level is [[Darboux's theorem (analysis)|Darboux's theorem]], that the derivative of a differentiable function satisfies the [[intermediate value property]]; weaker than continuity, but still a restricted class of functions. In more advanced analysis, regularity questions become central. The classical derivative turns out to be very inconvenient in many applications, such as to differential equations where many problems have a more natural [[calculus of variations|variational formulation]] using integration rather than differentiation. The functions one must consider are often not differentiable, and so generalizations of differentiability, and even what constitutes a function, are considered. Regularity then involves questions about whether the generalized derivatives are honest derivatives, and how smooth solutions to equations are. This is a common theme in measure theory, partial differential equations, Sobolev spaces, and the calculus of variations.{{sfn|Pugh|2015|loc=Chs. 3–4}}{{sfn|Folland|1999|loc=Chs. 3, 6, 9}}{{sfn|Royden|Fitzpatrick|2010|loc=Chs. 6–7}}


'''Definition.''' Let <math>(a_n)</math> be a real-valued sequence.  We say that <math>(a_n)</math> is a '''''Cauchy sequence''''' if, for any <math>\varepsilon > 0</math>, there exists a natural number <math>N</math> such that <math>m,n\geq N</math> implies that <math>|a_m-a_n| < \varepsilon</math>.
===Integration and measure===
Integration in analysis gives rigorous meaning to averaging, accumulation, and area.  The [[Riemann integral]] formalizes the integral using approximations by finite sums over intervals. The Riemann integral is put on a firm foundation in basic real analysis courses.  The [[fundamental theorem of calculus]], which relates integration and differentiation, is proved for the Riemann integral. The fundamental theorem underlies many exact computations in elementary integral calculus by reducing definite integrals to antiderivatives. The [[mean value theorem for integration]] states that the average value of a continuous function over an interval, defined by a Riemann integral, is equal to the value of the function at some point inside the interval. The mean value theorem is important in deriving analytic estimates in analytic applications; for example, one form of the Taylor remainder uses it.{{sfn|Abbott|2015|loc=Ch. 7}}{{sfn|Pugh|2015|loc=Ch. 4}}{{sfn|Rudin|1976|loc=Ch. 6}}


It can be shown that a real-valued sequence is Cauchy [[if and only if]] it is convergent. This property of the real numbers is expressed by saying that the real numbers endowed with the standard metric, <math>(\R, |\cdot|)</math>, is a '''''[[complete metric space]]'''''.  In a general metric space, however, a Cauchy sequence need not converge.
The [[Lebesgue integral]] is introduced in more advanced real analysis. Instead of approximating the integral by finite sums using partitions of the domain, the Lebesgue integral partitions the range instead. While the idea is similar to the Riemann integral, it differs from the Riemann integral in that one must formulate the area of horizontal slabs that are not connected rectangles, but may be distributed over complicated sets in the domain. Thus formulating the Lebesgue integral leads naturally to [[measure theory]], the fundamental question of which is how to measure subsets of the real line – that is, assign a notion of length – if the subsets are allowed to be very complicated.{{sfn|Royden|Fitzpatrick|2010|loc=Chs. 2–4}}{{sfn|Folland|1999|loc=Chs. 1–2}}


In addition, for real-valued sequences that are monotonic, it can be shown that the sequence is bounded if and only if it is convergent.
One reason for the success of the Lebesgue integral in analysis is that it is more compatible with limiting processes. For example, under relatively mild hypotheses, the limit of the Lebesgue integrals of members of a sequence of functions that converges only pointwise is equal to the integral of the pointwise limit ([[dominated convergence theorem]]).  Moreover, the Lebesgue integral admits a wider class of functions as integrable.  The Riemann integral, although not widely used in mathematical analysis, still has its uses however: most [[numerical integration]] relies on the Riemann rather than Lebesgue approach to integration.{{sfn|Royden|Fitzpatrick|2010|loc=Chs. 3–4}}{{sfn|Folland|1999|loc=Ch. 2}}


==== Uniform and pointwise convergence for sequences of functions ====
Measure theory treats length, area, volume, mass, and probability as instances of a general concept of measure. The theory allows one to identify functions that are equal except on a set of [[measure zero]], and one says that such functions are equal [[almost everywhere]].  In probability theory, the relevant notion is [[almost surely]]. An example is tossing a fair coin an infinite sequence of times. It is ''possible'' that the coin could come up tails every time, but this event has zero probability: it can be essentially excluded from any probabilistic argument.{{sfn|Royden|Fitzpatrick|2010|loc=Chs. 2–4}}{{sfn|Folland|1999|loc=Chs. 1–2}}
{{Main|Uniform convergence}}
In addition to sequences of numbers, one may also speak of ''sequences of functions'' ''on'' <math>E\subset \mathbb{R}</math>, that is, infinite, ordered families of functions <math>f_n:E\to\mathbb{R}</math>, denoted <math>(f_n)_{n=1}^\infty</math>, and their convergence properties.  However, in the case of sequences of functions, there are two kinds of convergence, known as ''pointwise convergence'' and ''uniform convergence'', that need to be distinguished.


Roughly speaking, pointwise convergence of functions <math>f_n</math> to a limiting function <math>f:E\to\mathbb{R}</math>, denoted <math>f_n \rightarrow f</math>, simply means that given any <math>x\in E</math>, <math>f_n(x)\to f(x)</math> as <math>n\to\infty</math>. In contrast, uniform convergence is a stronger type of convergence, in the sense that a uniformly convergent sequence of functions also converges pointwise, but not conversely. [[Uniform convergence]] requires members of the family of functions, <math>f_n</math>, to fall within some error <math>\varepsilon > 0</math> of <math>f</math> for ''every value of <math>x\in E</math>'', whenever <math>n\geq N</math>, for some integer <math>N</math>.  For a family of functions to uniformly converge, sometimes denoted <math>f_n\rightrightarrows f</math>, such a value of <math>N</math> must exist for any <math>\varepsilon>0</math> given, no matter how small. Intuitively, we can visualize this situation by imagining that, for a large enough <math>N</math>, the functions <math>f_N, f_{N+1}, f_{N+2},\ldots</math> are all confined within a 'tube' of width <math>2\varepsilon</math> about <math>f</math> (that is, between <math>f - \varepsilon</math> and <math>f+\varepsilon</math>) ''for every value in their domain'' <math>E</math>.
The [[monotone convergence theorem]], [[Fatou's lemma]], [[dominated convergence theorem]], and [[Fubini's theorem]] are basic theorems about the Lebesgue integral.{{sfn|Royden|Fitzpatrick|2010|loc=Chs. 2–4}}{{sfn|Folland|1999|loc=Chs. 1–2}} Measure-theoretic real analysis also supplies the language of probability and of many function spaces. In [[probability theory]], [[expected value]]s are integrals with respect to [[probability measure]]s. In the theory of [[Lp space|L<sup>''p''</sup> spaces]], functions are studied according to integrability properties and are identified when they agree almost everywhere.{{sfn|Royden|Fitzpatrick|2010|loc=Chs. 4, 6}}{{sfn|Folland|1999|loc=Chs. 2–3, 6}}


The distinction between pointwise and uniform convergence is important when exchanging the order of two limiting operations (e.g., taking a limit, a derivative, or integral) is desired: in order for the exchange to be well-behaved, many theorems of real analysis call for uniform convergence.  For example, a sequence of continuous functions (see [[Real analysis#Continuity|below]]) is guaranteed to converge to a continuous limiting function if the convergence is uniform, while the limiting function may not be continuous if convergence is only pointwise. [[Karl Weierstrass]] is generally credited for clearly defining the concept of uniform convergence and fully investigating its implications.
=== Series, function sequences, and representation ===
{{Main|Series (mathematics)|Power series|Fourier series|Fourier analysis}}
[[Infinite series]] are one of the basic limiting processes of real analysis. A numerical series is defined by the convergence of its [[partial sum]]s, and questions about its [[absolute convergence]], [[conditional convergence]], and [[Riemann series theorem|rearrangement]]s illustrate the difference between finite and infinite additions. [[Power series]] and [[Taylor series]] connect infinite series with differentiablity and analyticity.{{sfn|Abbott|2015|loc=Chs. 2, 6}}{{sfn|Rudin|1976|loc=Chs. 3, 8}}


=== Compactness ===
Sequences and series are used to approximate and represent functions. Uniform convergence gives conditions under which continuity, integrability, and sometimes differentiability, pass to the limit function. Power series can be used to represent some functions locally, while [[Fourier series]] represent periodic functions in terms of [[trigonometric series]]. These lead from elementary analysis to questions of [[approximation theory]] (how well a function is represented by a series or partial sum), [[harmonic analysis]] (what can be inferred about the regularity of the function from its representations), and the theory of function spaces (what classes of functions can be represented, and how well, in a certain way).{{sfn|Rudin|1976|loc=Chs. 7–8}}{{sfn|Folland|1999|loc=Chs. 8–9}}
{{Main|Compactness}}Compactness is a concept from [[general topology]] that plays an important role in many of the theorems of real analysis. The property of compactness is a generalization of the notion of a set being ''closed'' and ''bounded''. (In the context of real analysis, these notions are equivalent: a set in Euclidean space is compact if and only if it is closed and bounded.)  Briefly, a [[closed set]] contains all of its [[Boundary (topology)|boundary points]], while a set is [[Bounded set|bounded]] if there exists a real number such that the distance between any two points of the set is less than that number.  In <math>\mathbb{R}</math>, sets that are closed and bounded, and therefore compact, include the empty set, any finite number of points, [[Interval (mathematics)|closed intervals]], and their finite unions. However, this list is not exhaustive; for instance, the set <math>\{1/n:n\in\mathbb{N}\}\cup \{0}\</math> is a compact set; the [[Cantor set|Cantor ternary set]] <math>\mathcal{C}\subset [0,1]</math> is another example of a compact set. On the other hand, the set <math>\{1/n:n\in\mathbb{N}\}</math> is not compact because it is bounded but not closed, as the boundary point 0 is not a member of the set.  The set <math>[0,\infty)</math> is also not compact because it is closed but not bounded.


For subsets of the real numbers, there are several equivalent definitions of compactness.
=== Metric spaces and function spaces ===
Many of the theorems of real analysis can be formulated in [[metric space]]s. Metric spaces generalize the real line and [[Euclidean space]]s, in the sense that they have a notion of distance function defined on them that satisfies some natural rules. The basic theorems about limits and continuity carry through, essentially unchanged, in the more general setting, and so elementary treatments of the subject often describe this generalization.{{sfn|Rudin|1976|loc=Ch. 2}}{{sfn|Munkres|2000|loc=§§20–21}}  The generalizations to metric spaces are important in many areas of mathematics. One example of a metric space is a surface in Euclidean space. The analytic properties of metrics like this are similar to those of Euclidean space, and [[differential geometry]] involves the study of metric spaces like surfaces and their generalizations to higher dimensions.{{sfn|Lee|2018|loc=Chs. 2, 6}} Other metric spaces in mathematics are very different from Euclidean spaces. In [[number theory]] for example, metric spaces are used to encode arithmetic information and invariants, a basic example of which is the [[p-adic numbers]].{{sfn|Koblitz|1984|loc=Ch. I}}


'''Definition.''' A set <math>E\subset\mathbb{R}</math> is compact if it is closed and bounded.
In real analysis proper, most of the examples of metric spaces other than Euclidean spaces are [[function space]]s, that is spaces of functions that are defined by some property. In the basic examples considered in real analysis, these spaces of functions have a metric that comes from a [[norm (mathematics)|norm]]. An example is the space of continuous functions on the [[unit interval]], denoted <math>C([0,1])</math>.  Given a real-valued function on the unit interval, the maximum absolute value of the function defines the norm. The metric is the maximum absolute difference between two functions. The notion of convergence that this metric defines is that of uniform convergence, so that the language of uniform convergence of continuous functions can be packaged into a single metric space. This is an example of a [[Banach space]]: the metric that this norm defines is [[complete metric space|complete]], which is a consequence of a theorem that the uniform limit of continuous functions is continuous.{{sfn|Abbott|2015|loc=Ch. 6}}{{sfn|Rudin|1976|loc=Ch. 7}}{{sfn|Folland|1999|loc=Chs. 5–6}}


This definition also holds for Euclidean space of any finite dimension, <math>\mathbb{R}^n</math>, but it is not valid for metric spaces in general. The equivalence of the definition with the definition of compactness based on subcovers, given later in this section, is known as the [[Heine–Borel theorem|Heine-Borel theorem]].
Another example of metric spaces appearing in real analysis, where the metric comes from a norm, is the space <math>L^1</math> of Lebesgue integrable functions, and the distance function is the integral of the absolute value of the difference of two functions. (Here functions are regarded as the same if they differ on a set of measure zero.)  Thus, on a finite domain, the distance between two functions is essentially the average distance between their values. Spaces of [[square integrable function|square integrable]] measurable functions form a metric space, where the norm is induced by an [[inner product]]. This inner product is complete: the square integrable functions are a [[Hilbert space]].  The [[Lp space|L<sup>''p''</sup> spaces]] are generalizations of these. The metrics one gets from the L<sup>''p''</sup> spaces are complete: they are Banach spaces, with the usual identification of measurable functions that are equal almost everywhere.{{sfn|Royden|Fitzpatrick|2010|loc=Ch. 6}}{{sfn|Folland|1999|loc=Chs. 6–7}}


A more general definition that applies to all metric spaces uses the notion of a subsequence (see above).
These function spaces, while all being [[infinite dimensional]] [[vector space]]s, and share similar kinds of convergence theorems because they are all metric spaces, are different both globally as spaces, and locally in terms of the fine structure that functions must carry for membership. [[Functional analysis]] is concerned with these spaces and their generalizations, their global and convergence properties. [[Harmonic analysis]] is concerned with the fine properties needed for membership, and to estimate how operators determined by fine structure behave in theses spaces: a typical question here is to characterize when the derivative of a function is in a certain L<sup>''p''</sup> space, for example..{{sfn|Folland|1999|loc=Chs. 5–9}}{{sfn|Royden|Fitzpatrick|2010|loc=Chs. 6–7}}


'''Definition.''' A set <math>E</math> in a metric space is compact if every sequence in <math>E</math> has a convergent subsequence.
=== Completeness and compactness ===


This particular property is known as ''subsequential compactness''. In <math>\mathbb{R}</math>, a set is subsequentially compact if and only if it is closed and bounded, making this definition equivalent to the one given above.  Subsequential compactness is equivalent to the definition of compactness based on subcovers for metric spaces, but not for topological spaces in general.
Many applications of metric spaces to analysis rely on a metric space with extra properties that ensure that certain limits exist and belong to the space.  One such property is whether a metric space is [[complete metric space|complete]], and another is whether a metric space is [[compact metric space|compact]].{{sfn|Rudin|1976|loc=Ch. 2}}{{sfn|Pugh|2015|loc=Ch. 2}}{{sfn|Munkres|2000|loc=§§26–28}}


The most general definition of compactness relies on the notion of ''open covers'' and ''subcovers'', which is applicable to topological spaces (and thus to metric spaces and <math>\mathbb{R}</math> as special cases).  In brief, a collection of open sets <math>U_{\alpha}</math> is said to be an ''open cover'' of set <math>X</math> if the union of these sets is a superset of <math>X</math>.  This open cover is said to have a ''finite subcover'' if a finite subcollection of the <math>U_{\alpha}</math> could be found that also covers <math>X</math>.
Completeness of a space is formulated in terms of [[Cauchy sequence]]s within it.  A sequence in a space ''converges'' to a limit if all except finitely many of its members are arbitrarily close to the limit. Now, if a sequence converges, then all but finitely many of its members are close to the limit, and therefore must be close to one another. This property characterizes the Cauchy sequences in a metric space.  Stated a little more carefully, a sequence is Cauchy if, for any error tolerance, all but finitely many members lie within that error tolerance of each other.  A metric space is complete if every Cauchy sequence converges. The real numbers, for example, are complete in this sense.{{sfn|Abbott|2015|loc=Ch. 2}}{{sfn|Pugh|2015|loc=Ch. 2}}{{sfn|Rudin|1976|loc=Chs. 2–3}}


'''Definition.''' A set <math>X</math> in a topological space is compact if every open cover of <math>X</math> has a finite subcover.
One key theorem for complete metric spaces is the [[contraction mapping theorem]]. This theorem says that if a transformation <math>T</math> of a complete space decreases distances between points by at least a factor of <math>C<1</math>, so that <math>\operatorname{dist}(Tx,Ty)\le C\operatorname{dist}(x,y)</math>, then the transformation has a unique [[fixed point]], that is a point of the space such that <math>Tx=x</math>. This theorem is particularly important in [[differential equations]], because it supplies the basic existence and uniqueness theorem for solutions of ordinary differential equations – the [[Picard existence theorem]] – as well as a method for converging to it – [[Picard iteration]].  Furthermore, it provides estimates of the rate of convergence, provided one can determine the contractivity constant <math>C</math>.{{sfn|Pugh|2015|loc=Ch. 2}}{{sfn|Folland|1999|loc=Ch. 5}}


Compact sets are well-behaved with respect to properties like convergence and continuity. For instance, any Cauchy sequence in a compact metric space is convergent. As another example, the image of a compact metric space under a continuous map is also compact.
Another notion in metric spaces is compactness. A metric space is compact if every sequence has a convergent subsequence. Compact metric spaces are also complete, but compactness is useful for guaranteeing the existence of limits without needing to study a separate Cauchy condition.  In basic real analysis, the [[Bolzano–Weierstrass theorem]] shows that a subset of Euclidean space is compact if and only if it is closed and bounded.{{sfn|Abbott|2015|loc=Ch. 3}}{{sfn|Pugh|2015|loc=Ch. 2}}{{sfn|Rudin|1976|loc=Chs. 2–3}}


===Continuity===
A more general notion of compactness applies to general [[topological space]]s that are not necessarily metric spaces, and it is based on the idea that compact spaces should generalized finite sets.  Finite sets have the property that whenever they are covered by a family of subsets, so that every member of the set belongs to some member of the cover, it is possible to cover the finite set with a finite subfamily. For example, one can pick from each element of the finite set one member of the covering family that contains it.  A metric space is compact in this sense if every covering by open sets has a finite subcover. An example of a compact set in this sense is a closed interval <math>[a,b]</math>: whenever this set is covered by open intervals, it is possible to cover it with finitely many of the intervals. This can be shown by a straightforward application of the least upper bound property. A compact set in the real line can be shown to be bounded, by considering an open cover by intervals <math>(-n,n)</math> and using a finite subcover to extract a bound.  It can also be shown to be closed by a simple proof by contradiction. Another foundational result in real analysis is the [[Heine–Borel theorem]], which states the converse to this: a subset of the real line (or Euclidean space, more generally) is compact in this sense ''if and only if'' it is closed and bounded.{{sfn|Pugh|2015|loc=Ch. 2}}{{sfn|Rudin|1976|loc=Ch. 2}}{{sfn|Munkres|2000|loc=§§26–28}}
{{Main|Continuous function}}
A [[function (mathematics)|function]] from the set of [[real number]]s to the real numbers can be represented by a [[graph of a function|graph]] in the [[Cartesian coordinate system|Cartesian plane]]; such a function is continuous if, roughly speaking, the graph is a single unbroken [[curve]] with no "holes" or "jumps".


There are several ways to make this intuition mathematically rigorous. Several definitions of varying levels of generality can be given. In cases where two or more definitions are applicable, they are readily shown to be [[Equivalence relation|equivalent]] to one another, so the most convenient definition can be used to determine whether a given function is continuous or not. In the first definition given below, <math>f:I\to\R</math> is a function defined on a non-degenerate interval <math>I</math> of the set of real numbers as its domain. Some possibilities include <math>I=\R</math>, the whole set of real numbers, an [[open interval]] <math>I = (a, b) = \{x \in \R \mid a < x < b \}, </math> or a [[closed interval]] <math>I = [a, b] = \{x \in \R \mid a \leq x \leq b\}. </math> Here, <math>a</math> and <math>b</math> are distinct real numbers, and we exclude the case of <math>I</math> being empty or consisting of only one point, in particular.
Compactness plays an important role through its interaction with continuity. The [[extreme value theorem]] in calculus, for example, says that a continuous function on a closed interval <math>[a,b]</math> attains both a maximum and minimum value. In real analysis, this generalizes: any continuous real-valued function on a compact metric space attains both a maximum and minimum value. This result is important in [[optimization theory]] because it guarantees the existence of maxima and minima, often in very general situations where compactness is present but one is no longer working on subsets of the real line (or other Euclidean space).{{sfn|Abbott|2015|loc=Chs. 3–4}}{{sfn|Pugh|2015|loc=Ch. 2}}{{sfn|Rudin|1976|loc=Ch. 4}}


'''Definition.'''  If <math>I\subset \mathbb{R}</math> is a non-degenerate interval, we say that <math>f:I \to \R</math> is '''''continuous at''''' <math>p\in I</math> if <math display="inline">\lim_{x \to p} f(x) = f(p)</math>.  We say that <math>f</math> is a '''''continuous map''''' if <math>f</math> is continuous at every <math>p\in I</math>.
Another theorem in real analysis, relating to compactness and continuous functions, is the [[Arzelà–Ascoli theorem]]. It asserts that if a sequence of continuous functions on a compact metric space is uniformly bounded (all functions share a common upper and lower bound) and [[equicontinuous]] (all functions share a common modulus of continuity), then the sequence of functions has a uniformly convergent subsequence. This theorem has applications to differential equations, because it can be used to guarantee existence of solutions to differential equations where the stricter criteria of the Picard existence theorem do not hold.  An example application is the [[Peano existence theorem]].  The Arzelà–Ascoli theorem is itself a kind of compactness assertion, characterizing compactness in the Banach space of functions on a compact metric space.{{sfn|Pugh|2015|loc=Ch. 2}}{{sfn|Rudin|1976|loc=Ch. 7}}{{sfn|Folland|1999|loc=Chs. 5–6}}


In contrast to the requirements for <math>f</math> to have a limit at a point <math>p</math>, which do not constrain the behavior of <math>f</math> at <math>p</math> itself, the following two conditions, in addition to the existence of <math display="inline">\lim_{x\to p} f(x)</math>, must also hold in order for <math>f</math> to be continuous at <math>p</math>: '''(i)''' <math>f</math> must be defined at <math>p</math>, i.e., <math>p</math> is in the domain of <math>f</math>; ''and'' '''(ii)''' <math>f(x)\to f(p)</math> as <math>x\to p</math>.  The definition above actually applies to any domain <math>E</math> that does not contain an [[isolated point]], or equivalently, <math>E</math> where every <math>p\in E</math> is a [[limit point]] of <math>E</math>. A more general definition applying to <math>f:X\to\mathbb{R}</math> with a general domain <math>X\subset \mathbb{R}</math> is the following:
==Important results==
Important elementary results put the techniques of calculus on rigorous footing.  They include the [[Bolzano–Weierstrass theorem|Bolzano–Weierstrass]] and [[Heine–Borel theorem]]s, [[L'Hopital's rule]], the [[mean value theorem]], [[Taylor's theorem]], the [[fundamental theorem of calculus]], and the [[extreme value theorem]].  


'''Definition.'''  If <math>X</math> is an arbitrary subset of <math>\mathbb{R}</math>, we say that <math>f:X\to\mathbb{R}</math> is '''''continuous at''''' <math>p\in X</math> if, for any <math>\varepsilon>0</math>, there exists <math>\delta>0</math> such that for all <math>x\in X</math>, <math>|x-p|<\delta</math> implies that <math>|f(x)-f(p)| < \varepsilon</math>. We say that <math>f</math> is a '''''continuous map''''' if <math>f</math> is continuous at every <math>p\in X</math>.
Other results, also taught in an elementary course, are motivated by applications, chiefly to [[differential equations]], [[multivariable calculus]], and [[Fourier analysis]] which are part of the undergraduate mathematics curriculum.  They are the [[Arzelà–Ascoli theorem|Arzelà-Ascoli theorem]], the [[Stone–Weierstrass theorem|Stone-Weierstrass theorem]], the [[Banach fixed-point theorem]], the [[inverse function theorem|inverse]] and [[implicit function theorem]]s, and [[Stokes' theorem]].{{sfn|Abbott|2015|loc=Chs. 2–7}}{{sfn|Pugh|2015|loc=Chs. 1–5}}{{sfn|Rudin|1976|loc=Chs. 2–7}}


A consequence of this definition is that <math>f</math> is ''trivially continuous at any isolated point'' <math>p\in X</math>.  This somewhat unintuitive treatment of isolated points is necessary to ensure that our definition of continuity for functions on the real line is consistent with the most general definition of continuity for maps between [[topological space]]s (which includes [[metric space]]s and <math>\mathbb{R}</math> in particular as special cases).  This definition, which extends beyond the scope of our discussion of real analysis, is given below for completeness.
More advanced graduate-level courses also cover the basics of [[measure theory]], the basic theorems of which are: [[Egorov's theorem]], [[Lusin's theorem]], [[Fatou's lemma]], the [[Monotone convergence theorem|monotone]] and [[dominated convergence theorem]]s, and [[Fubini's theorem]].{{sfn|Royden|Fitzpatrick|2010|loc=Chs. 2–7}}{{sfn|Folland|1999|loc=Chs. 1–6}}


'''Definition.'''  If <math>X</math> and <math>Y</math> are topological spaces, we say that <math>f:X\to Y</math> is '''''continuous at''''' <math>p\in X</math> if <math>f^{-1} (V)</math> is a [[Neighborhood (topology)|neighborhood]] of <math>p</math> in <math>X</math> for every neighborhood <math>V</math> of <math>f(p)</math> in <math>Y</math>.  We say that <math>f</math> is a '''''continuous map''''' if <math>f^{-1}(U)</math> is open in <math>X</math> for every <math>U</math> open in <math>Y</math>.
Other advanced results are sometimes also covered at the graduate level, because these are motivated by applications to other areas in the graduate curriculum. They include the [[Radon–Nikodym theorem]], [[Lebesgue decomposition theorem]], and [[Riesz representation theorem]]. Sometimes results such as the [[Lebesgue differentiation theorem]] are also included, as applications of [[harmonic analysis]] to questions of real analysis.


(Here, <math>f^{-1}(S)</math> refers to the [[preimage]] of <math>S\subset Y</math> under <math>f</math>.)
== Generalizations and related areas of mathematics ==
 
Various ideas from real analysis can be generalized from the real line to broader or more abstract contexts.  These generalizations link real analysis to other disciplines and subdisciplines. For instance, generalization of ideas like continuous functions and compactness from real analysis to [[metric space]]s and [[topological space]]s connects real analysis to the field of [[general topology]], while generalization of finite-dimensional Euclidean spaces to infinite-dimensional analogs led to the concepts of [[Banach space]]s and [[Hilbert space]]s and, more generally to [[functional analysis]].  [[Georg Cantor]]'s investigation of sets and sequence of real numbers, mappings between them, and the foundational issues of real analysis gave birth to [[naive set theory]].  The study of issues of [[Limit (mathematics)|convergence]] for sequences of functions eventually gave rise to [[Fourier analysis]] as a subdiscipline of mathematical analysis.   Investigation of the consequences of generalizing differentiability from functions of a real variable to ones of a complex variable gave rise to the concept of [[holomorphic function]]s and the inception of [[complex analysis]] as another distinct subdiscipline of analysis.  On the other hand, the generalization of integration from the Riemann sense to that of Lebesgue led to the formulation of the concept of abstract [[measure space]]s, a fundamental concept in [[measure theory]].  Finally, the generalization of integration from the real line to curves and surfaces in higher dimensional space brought about the study of [[vector calculus]], whose further generalization and formalization played an important role in the evolution of the concepts of [[differential form]]s and [[Differentiable manifold|smooth (differentiable) manifolds]] in [[differential geometry]] and other closely related areas of [[geometry]] and [[topology]].{{cn|date=June 2026}}
====Uniform continuity====
{{Main|Uniform continuity}}
'''Definition.''' If <math>X</math> is a subset of the [[real number]]s, we say a function <math>f:X\to\mathbb{R}</math> is '''''uniformly continuous''''' '''''on''''' <math>X</math> if, for any <math>\varepsilon > 0</math>, there exists a <math>\delta>0</math> such that for all <math>x,y\in X</math>, <math>|x-y|<\delta</math> implies that <math>|f(x)-f(y)| < \varepsilon</math>.
 
Explicitly, when a function is uniformly continuous on <math>X</math>, the choice of <math>\delta</math> needed to fulfill the definition must work for ''all of'' <math>X</math> for a given <math>\varepsilon</math>.  In contrast, when a function is continuous at every point <math>p\in X</math> (or said to be continuous on <math>X</math>), the choice of <math>\delta</math> may depend on both <math>\varepsilon</math> ''and'' <math>p</math>. In contrast to simple continuity, uniform continuity is a property of a function that only makes sense with a specified domain; to speak of uniform continuity at a single point <math>p</math> is meaningless.
 
On a compact set, it is easily shown that all continuous functions are uniformly continuous.  If <math>E</math> is a bounded noncompact subset of <math>\mathbb{R}</math>, then there exists <math>f:E\to\mathbb{R}</math> that is continuous but not uniformly continuous.  As a simple example, consider <math>f:(0,1)\to\mathbb{R}</math> defined by <math>f(x)=1/x</math>.  By choosing points close to 0, we can always make <math>|f(x)-f(y)| > \varepsilon</math> for any single choice of <math>\delta>0</math>, for a given <math>\varepsilon > 0</math>.
 
====Absolute continuity====
{{Main|Absolute continuity}}
'''Definition.''' Let <math>I\subset\mathbb{R}</math> be an [[interval (mathematics)|interval]] on the [[real line]]. A function <math>f:I \to \mathbb{R}</math> is said to be '''''absolutely continuous''''' '''''on''''' <math>I</math> if for every positive number <math>\varepsilon</math>, there is a positive number <math>\delta</math> such that whenever a finite sequence of [[pairwise disjoint]] sub-intervals <math>(x_1, y_1), (x_2,y_2),\ldots, (x_n,y_n)</math> of <math>I</math> satisfies<ref>{{harvnb|Royden|1988|loc=Sect. 5.4, page 108}}; {{harvnb|Nielsen|1997|loc=Definition 15.6 on page 251}}; {{harvnb|Athreya|Lahiri|2006|loc=Definitions 4.4.1, 4.4.2 on pages 128,129}}. The interval ''I'' is assumed to be bounded and closed in the former two books but not the latter book.</ref>
:<math>\sum_{k=1}^{n} (y_k - x_k) < \delta</math>
then
:<math>\sum_{k=1}^{n} | f(y_k) - f(x_k) | < \varepsilon.</math>
Absolutely continuous functions are continuous: consider the case ''n'' = 1 in this definition.  The collection of all absolutely continuous functions on ''I'' is denoted AC(''I'').  Absolute continuity is a fundamental concept in the Lebesgue theory of integration, allowing the formulation of a generalized version of the fundamental theorem of calculus that applies to the Lebesgue integral.
 
===Differentiation===
{{Main|Derivative|Differential calculus|}}
The notion of the ''derivative'' of a function or ''differentiability'' originates from the concept of approximating a function near a given point using the "best" linear approximation.  This approximation, if it exists, is unique and is given by the line that is tangent to the function at the given point <math>a</math>, and the slope of the line is the derivative of the function at <math>a</math>.
 
A function <math>f:\mathbb{R}\to\mathbb{R}</math> is '''''differentiable at <math>a</math>''''' if the [[limit of a function|limit]]
 
:<math>f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}</math>
 
exists.  This limit is known as the '''''derivative of <math>f</math> at <math>a</math>''''', and the function <math>f'</math>, possibly defined on only a subset of <math>\mathbb{R}</math>, is the '''''derivative''''' (or '''''derivative function''''') '''''of''''' '''''<math>f</math>'''''.  If the derivative exists everywhere, the function is said to be '''''differentiable'''''.
 
As a simple consequence of the definition, <math>f</math> is continuous at '''''<math>a</math>''''' if it is differentiable there.  Differentiability is therefore a stronger regularity condition (condition describing the "smoothness" of a function) than continuity, and it is possible for a function to be continuous on the entire real line but not differentiable anywhere (see [[Weierstrass function|Weierstrass's nowhere differentiable continuous function]]).  It is possible to discuss the existence of higher-order derivatives as well, by finding the derivative of a derivative function, and so on.
 
One can classify functions by their '''''differentiability class'''''. The class <math>C^0</math> (sometimes <math>C^0([a,b])</math> to indicate the interval of applicability) consists of all continuous functions.  The class <math>C^1</math> consists of all [[differentiable function]]s whose derivative is continuous; such functions are called '''''continuously differentiable'''''.  Thus, a <math>C^1</math> function is exactly a function whose derivative exists and is of class <math>C^0</math>.  In general, the classes ''<math>C^k</math>'' can be defined [[recursion|recursively]] by declaring <math>C^0</math> to be the set of all continuous functions and declaring ''<math>C^k</math>'' for any positive integer <math>k</math> to be the set of all differentiable functions whose derivative is in <math>C^{k-1}</math>.  In particular, ''<math>C^k</math>'' is contained in <math>C^{k-1}</math> for every <math>k</math>, and there are examples to show that this containment is strict.  Class <math>C^\infty</math> is the intersection of the sets ''<math>C^k</math>'' as ''<math>k</math>'' varies over the non-negative integers, and the members of this class are known as the '''''smooth functions'''''.  Class <math>C^\omega</math> consists of all [[analytic function]]s, and is strictly contained in <math>C^\infty</math> (see [[bump function]] for a smooth function that is not analytic).
 
===Series===
{{Main|Series (mathematics)}}
A series formalizes the imprecise notion of taking the sum of an endless sequence of numbers.  The idea that taking the sum of an "infinite" number of terms can lead to a finite result was counterintuitive to the ancient Greeks and led to the formulation of a number of paradoxes by Zeno and other philosophers.  The modern notion of assigning a value to a series avoids dealing with the ill-defined notion of adding an "infinite" number of terms.  Instead, the finite sum of the first <math>n</math> terms of the sequence, known as a partial sum, is considered, and the concept of a limit is applied to the sequence of partial sums as <math>n</math> grows without bound.  The series is assigned the value of this limit, if it exists.
 
Given an (infinite) [[sequence]] <math>(a_n)</math>, we can define an associated '''''series''''' as the formal mathematical object {{nowrap|<math display="inline">a_1 + a_2 + a_3 + \cdots = \sum_{n=1}^{\infty} a_n</math>,}} sometimes simply written as <math display="inline">\sum a_n</math>.  The '''''partial sums''''' of a series <math display="inline">\sum a_n</math> are the numbers <math display="inline">s_n=\sum_{j=1}^n a_j</math>.  A series <math display="inline">\sum a_n</math> is said to be '''''convergent''''' if the sequence consisting of its partial sums, <math>(s_n)</math>, is convergent; otherwise it is '''''divergent'''''.  The '''''sum''''' of a convergent series is defined as the number {{nowrap|<math display="inline">s = \lim_{n \to \infty} s_n</math>.}}
 
The word "sum" is used here in a metaphorical sense as a shorthand for taking the limit of a sequence of partial sums and should not be interpreted as simply "adding" an infinite number of terms.  For instance, in contrast to the behavior of finite sums, rearranging the terms of an infinite series may result in convergence to a different number (see the article on the ''[[Riemann rearrangement theorem]]'' for further discussion).
 
An example of a convergent series is a [[geometric series]] which forms the basis of one of Zeno's famous [[Zeno's paradoxes|paradoxes]]:
 
:<math>\sum_{n=1}^\infty \frac{1}{2^n} = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots = 1 .</math>
 
In contrast, the [[Harmonic series (mathematics)|harmonic series]] has been known since the Middle Ages to be a divergent series:
 
:<math>\sum_{n=1}^\infty \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \cdots = \infty .</math>
 
(Here, "<math>=\infty</math>" is merely a notational convention to indicate that the partial sums of the series grow without bound.)
 
A series <math display="inline">\sum a_n</math> is said to '''''[[Absolute convergence|converge absolutely]]''''' if <math display="inline">\sum |a_n|</math> is convergent.  A convergent series <math display="inline">\sum a_n</math> for which <math display="inline">\sum |a_n|</math> diverges is said to '''''converge''''' '''''non-absolutely'''''.<ref>The term '''''unconditional convergence''''' refers to series whose sum does not depend on the order of the terms (i.e., any rearrangement gives the same sum).  Convergence is termed '''''conditional''''' otherwise.  For series in <math>\R^n</math>, it can be shown that absolute convergence and unconditional convergence are equivalent.  Hence, the term "conditional convergence" is often used to mean non-absolute convergence.  However, in the general setting of Banach spaces, the terms do not coincide, and there are unconditionally convergent series that do not converge absolutely.</ref>  It is easily shown that absolute convergence of a series implies its convergence.  On the other hand, an example of a series that converges non-absolutely is
 
:<math>\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots = \ln 2 .</math>
 
====Taylor series====
{{Main|Taylor series}}
The Taylor series of a [[real-valued function|real]] or [[complex-valued function]] ''ƒ''(''x'') that is [[infinitely differentiable function|infinitely differentiable]] at a [[real number|real]] or [[complex number]] ''a'' is the [[power series]]
<!--
As stated below, the Taylor series need not equal the function. So please don't write f(x)=... here. In other words,
 
DO NOT CHANGE ANYTHING ABOUT THIS FORMULA-->:<math>f(a) + \frac{f'(a)}{1!} (x-a) + \frac{f''(a)}{2!} (x-a)^2 + \frac{f^{(3)}(a)}{3!} (x-a)^3 + \cdots. </math><!---->
 
which can be written in the more compact [[Summation#Capital-sigma notation|sigma notation]] as
 
:<math> \sum_{n=0} ^ {\infty} \frac {f^{(n)}(a)}{n!} \, (x-a)^{n}</math>
 
where ''n''! denotes the [[factorial]] of ''n'' and ''ƒ''<sup>&nbsp;(''n'')</sup>(''a'') denotes the ''n''th [[derivative]] of ''ƒ'' evaluated at the point ''a''. The derivative of order zero ''ƒ'' is defined to be ''ƒ'' itself and {{nowrap|(''x'' − ''a'')<sup>0</sup>}} and 0! are both defined to be&nbsp;1. In the case that {{nowrap|''a'' {{=}} 0}}, the series is also called a Maclaurin series.
 
A Taylor series of ''f'' about point ''a'' may diverge, converge at only the point ''a'', converge for all ''x'' such that <math>|x-a|<R</math> (the largest such ''R'' for which convergence is guaranteed is called the ''radius of convergence''), or converge on the entire real line. Even a converging Taylor series may converge to a value different from the value of the function at that point.  If the Taylor series at a point has a nonzero [[radius of convergence]], and sums to the function in the [[disc of convergence]], then the function is [[analytic function|analytic]]. The analytic functions have many fundamental properties. In particular, an analytic function of a real variable extends naturally to a function of a complex variable. It is in this way that the [[exponential function]], the [[logarithm]], the [[trigonometric functions]] and their [[inverse trigonometric functions|inverses]] are extended to functions of a complex variable.
 
====Fourier series====
{{Main|Fourier series}}
[[Image:Fourier Series.svg|thumb|200px|The first four partial sums of the [[Fourier series]] for a [[Square wave (waveform)|square wave]]. Fourier series are an important tool in real analysis.]]
Fourier series decomposes [[periodic function]]s or periodic signals into the sum of a (possibly infinite) set of simple oscillating functions, namely [[sine wave|sines and cosines]]  (or [[complex exponential]]s). The study of Fourier series typically occurs and is handled within the branch [[mathematics]] > [[mathematical analysis]] > [[Fourier analysis]].
 
===Integration===
Integration is a formalization of the problem of finding the area bound by a curve and the related problems of determining the length of a curve or volume enclosed by a surface.  The basic strategy to solving problems of this type was known to the ancient Greeks and Chinese, and was known as the ''[[method of exhaustion]]''. Generally speaking, the desired area is bounded from above and below, respectively, by increasingly accurate circumscribing and inscribing polygonal approximations whose exact areas can be computed.  By considering approximations consisting of a larger and larger ("infinite") number of smaller and smaller ("infinitesimal") pieces, the area bound by the curve can be deduced, as the upper and lower bounds defined by the approximations converge around a common value.
 
The spirit of this basic strategy can easily be seen in the definition of the Riemann integral, in which the integral is said to exist if upper and lower Riemann (or Darboux) sums converge to a common value as thinner and thinner rectangular slices ("refinements") are considered.  Though the machinery used to define it is much more elaborate compared to the Riemann integral, the Lebesgue integral was defined with similar basic ideas in mind.  Compared to the Riemann integral, the more sophisticated Lebesgue integral allows area (or length, volume, etc.; termed a "measure" in general) to be defined and computed for much more complicated and irregular subsets of Euclidean space, although there still exist "non-measurable" subsets for which an area cannot be assigned.
 
====Riemann integration====
{{Main|Riemann integral}}
The Riemann integral is defined in terms of [[Riemann sum]]s of functions with respect to tagged partitions of an interval. Let <math>[a,b]</math> be a [[Interval (mathematics)|closed interval]] of the real line; then a '''''tagged partition''''' <math>\cal{P}</math> of <math>[a,b]</math> is a finite sequence
 
:<math> a = x_0 \le t_1 \le x_1 \le t_2 \le x_2 \le \cdots \le x_{n-1} \le t_n \le x_n = b . \,\!</math>
 
This partitions the interval <math>[a,b]</math> into <math>n</math> sub-intervals <math>[x_{i-1},x_i]</math> indexed by <math>i=1,\ldots, n</math>, each of which is "tagged" with a distinguished point <math>t_i\in[x_{i-1},x_i]</math>. For a function <math>f</math> bounded on <math>[a,b]</math>, we define the '''''Riemann sum''''' of <math>f</math> with respect to tagged partition <math>\cal{P}</math> as
 
:<math>\sum_{i=1}^{n} f(t_i) \Delta_i, </math>
 
where <math>\Delta_i=x_i-x_{i-1}</math> is the width of sub-interval <math>i</math>.  Thus, each term of the sum is the area of a rectangle with height equal to the function value at the distinguished point of the given sub-interval, and width the same as the sub-interval width. The '''''mesh''''' of such a tagged partition is the width of the largest sub-interval formed by the partition, <math display="inline">\|\Delta_i\| = \max_{i=1,\ldots, n}\Delta_i</math>. We say that the '''''Riemann integral''''' of <math>f</math> on <math>[a,b]</math> is <math>S</math> if for any <math>\varepsilon>0</math> there exists <math>\delta>0</math> such that, for any tagged partition <math>\cal{P}</math> with mesh <math>\| \Delta_i \| < \delta</math>, we have
 
::<math>\left| S - \sum_{i=1}^{n} f(t_i)\Delta_i \right| < \varepsilon.</math>
 
This is sometimes denoted <math display="inline">\mathcal{R}\int_{a}^b f=S</math>.  When the chosen tags give the maximum (respectively, minimum) value of each interval, the Riemann sum is known as the upper (respectively, lower) '''''Darboux sum'''''. A function is '''''Darboux integrable''''' if the upper and lower [[Darboux integral|Darboux sums]] can be made to be arbitrarily close to each other for a sufficiently small mesh.  Although this definition gives the Darboux integral the appearance of being a special case of the Riemann integral, they are, in fact, equivalent, in the sense that a function is Darboux integrable if and only if it is Riemann integrable, and the values of the integrals are equal.  In fact, calculus and real analysis textbooks often conflate the two, introducing the definition of the Darboux integral as that of the Riemann integral, due to the slightly easier to apply definition of the former.
 
The [[fundamental theorem of calculus]] asserts that integration and differentiation are inverse operations in a certain sense.
 
====Lebesgue integration and measure====
{{Main|Lebesgue integral}}
'''Lebesgue integration''' is a mathematical construction that extends the integral to a larger class of functions; it also extends the [[domain of a function|domains]] on which these functions can be defined.  The concept of a [[Measure (mathematics)|'''''measure''''']], an abstraction of length, area, or volume, is central to Lebesgue integral [[probability theory]].


===Distributions===
===Distributions===
Line 219: Line 92:


Techniques from the [[theory of analytic functions]] of a complex variable are often used in real analysis – such as evaluation of real integrals by [[residue theorem|residue calculus]].
Techniques from the [[theory of analytic functions]] of a complex variable are often used in real analysis – such as evaluation of real integrals by [[residue theorem|residue calculus]].
==Important results==
Important results include the [[Bolzano–Weierstrass theorem|Bolzano–Weierstrass]] and [[Heine–Borel theorem]]s, the [[intermediate value theorem]] and [[mean value theorem]], [[Taylor's theorem]], the [[fundamental theorem of calculus]], the [[Arzelà–Ascoli theorem|Arzelà-Ascoli theorem]], the [[Stone–Weierstrass theorem|Stone-Weierstrass theorem]], [[Fatou's lemma]], and the [[Monotone convergence theorem|monotone convergence]] and [[dominated convergence theorem]]s.
== Generalizations and related areas of mathematics ==
Various ideas from real analysis can be generalized from the real line to broader or more abstract contexts.  These generalizations link real analysis to other disciplines and subdisciplines. For instance, generalization of ideas like continuous functions and compactness from real analysis to [[metric space]]s and [[topological space]]s connects real analysis to the field of [[general topology]], while generalization of finite-dimensional Euclidean spaces to infinite-dimensional analogs led to the concepts of [[Banach space]]s and [[Hilbert space]]s and, more generally to [[functional analysis]].  [[Georg Cantor]]'s investigation of sets and sequence of real numbers, mappings between them, and the foundational issues of real analysis gave birth to [[naive set theory]].  The study of issues of [[Limit (mathematics)|convergence]] for sequences of functions eventually gave rise to [[Fourier analysis]] as a subdiscipline of mathematical analysis.  Investigation of the consequences of generalizing differentiability from functions of a real variable to ones of a complex variable gave rise to the concept of [[holomorphic function]]s and the inception of [[complex analysis]] as another distinct subdiscipline of analysis.  On the other hand, the generalization of integration from the Riemann sense to that of Lebesgue led to the formulation of the concept of abstract [[measure space]]s, a fundamental concept in [[measure theory]].  Finally, the generalization of integration from the real line to curves and surfaces in higher dimensional space brought about the study of [[vector calculus]], whose further generalization and formalization played an important role in the evolution of the concepts of [[differential form]]s and [[Differentiable manifold|smooth (differentiable) manifolds]] in [[differential geometry]] and other closely related areas of [[geometry]] and [[topology]].


==See also==
==See also==
Line 232: Line 99:
* [[Real coordinate space]]
* [[Real coordinate space]]
* [[Complex analysis]]
* [[Complex analysis]]
* [[Mathematical analysis]]


==References==
==References==
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==Sources==
==Sources==
* {{citation | last1=Athreya | first1=Krishna B. | last2=Lahiri | first2=Soumendra N. | title = Measure theory and probability theory | publisher = Springer | year = 2006 | isbn=0-387-32903-X }}
* {{citation |last=Abbott |first=Stephen |title=Understanding Analysis |edition=2nd |series=Undergraduate Texts in Mathematics |publisher=Springer |location=New York |year=2015 |isbn=978-1-4939-2711-1 |doi=10.1007/978-1-4939-2712-8}}.
* {{citation | last=Nielsen | first=Ole A. | title = An introduction to integration and measure theory | publisher = Wiley-Interscience | year = 1997 | isbn=0-471-59518-7 }}
* {{citation | last1=Athreya | first1=Krishna B. | last2=Lahiri | first2=Soumendra N. | title = Measure theory and probability theory | publisher = Springer | year = 2006 | isbn=0-387-32903-X }}.
* {{citation | last=Royden | first=H.L. | title = Real Analysis | publisher = Collier Macmillan | edition=third| year = 1988 | isbn=0-02-404151-3 }}
* {{citation |last1=Bartle |first1=Robert G. |last2=Sherbert |first2=Donald R. |title=Introduction to Real Analysis |edition=4th |publisher=John Wiley & Sons |location=New York |year=2011 |isbn=978-0-471-43331-6}}.
* {{citation |last=Folland |first=Gerald B. |title=Real Analysis: Modern Techniques and Their Applications |edition=2nd |series=Pure and Applied Mathematics |publisher=Wiley |location=New York |year=1999 |isbn=978-0-471-31716-6}}.
* {{citation |last=Koblitz |first=Neal |title=p-adic Numbers, p-adic Analysis, and Zeta-Functions |edition=2nd |series=Graduate Texts in Mathematics |volume=58 |publisher=Springer |location=New York |year=1984 |isbn=978-0-387-96017-3 |doi=10.1007/978-1-4612-1112-9}}.
* {{citation |last=Lee |first=John M. |title=Introduction to Riemannian Manifolds |edition=2nd |series=Graduate Texts in Mathematics |volume=176 |publisher=Springer |location=Cham |year=2018 |isbn=978-3-319-91755-9 |doi=10.1007/978-3-319-91755-9}}.
* {{citation |last=Munkres |first=James R. |title=Topology |edition=2nd |publisher=Prentice Hall |location=Upper Saddle River, New Jersey |year=2000 |isbn=0-13-181629-2}}.
* {{citation | last=Nielsen | first=Ole A. | title = An introduction to integration and measure theory | publisher = Wiley-Interscience | year = 1997 | isbn=0-471-59518-7 }}.
* {{citation |last=Pugh |first=Charles Chapman |title=Real Mathematical Analysis |edition=2nd |series=Undergraduate Texts in Mathematics |publisher=Springer |location=Cham |year=2015 |isbn=978-3-319-17770-0 |doi=10.1007/978-3-319-17771-7}}.
* {{citation |last1=Royden |first1=H. L. |last2=Fitzpatrick |first2=Patrick M. |title=Real Analysis |edition=4th |publisher=Pearson |location=Boston |year=2010 |isbn=978-0-13-143747-0}}.


==Bibliography==
==Bibliography==
Line 255: Line 130:


==External links==
==External links==
* [https://textbooks.opensuny.org/how-we-got-from-there-to-here-a-story-of-real-analysis/ How We Got From There to Here: A Story of Real Analysis] by Robert Rogers and Eugene Boman
* [https://textbooks.opensuny.org/how-we-got-from-there-to-here-a-story-of-real-analysis/ How We Got From There to Here: A Story of Real Analysis] {{Webarchive|url=https://web.archive.org/web/20190222193438/https://textbooks.opensuny.org/how-we-got-from-there-to-here-a-story-of-real-analysis/ |date=2019-02-22 }} by Robert Rogers and Eugene Boman
* [http://www.worldscientific.com/worldscibooks/10.1142/8580 A First Course in Analysis] by Donald Yau
* [http://www.worldscientific.com/worldscibooks/10.1142/8580 A First Course in Analysis] by Donald Yau
* [http://www.analysiswebnotes.com Analysis WebNotes] by John Lindsay Orr
* [http://www.analysiswebnotes.com Analysis WebNotes] by John Lindsay Orr
* [http://www.mathcs.org/analysis/reals/index.html Interactive Real Analysis] by  Bert G. Wachsmuth
* [http://www.mathcs.org/analysis/reals/index.html Interactive Real Analysis] by  Bert G. Wachsmuth
* [http://www-groups.mcs.st-andrews.ac.uk/~john/analysis/index.html A First Analysis Course] by John O'Connor
* [http://www-groups.mcs.st-andrews.ac.uk/~john/analysis/index.html A First Analysis Course] {{Webarchive|url=https://web.archive.org/web/20070927235711/http://www-groups.mcs.st-andrews.ac.uk/~john/analysis/index.html |date=2007-09-27 }} by John O'Connor
* [http://www.trillia.com/zakon-analysisI.html Mathematical Analysis I] by Elias Zakon
* [http://www.trillia.com/zakon-analysisI.html Mathematical Analysis I] by Elias Zakon
* [http://www.trillia.com/zakon-analysisII.html Mathematical Analysis II] by Elias Zakon
* [http://www.trillia.com/zakon-analysisII.html Mathematical Analysis II] by Elias Zakon

Latest revision as of 05:16, 23 June 2026

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File:Illustration of supremum.svg
The least-upper-bound property (completeness) of the real numbers: that every non-empty bounded above subset has a supremum. Virtually every major theorem in real analysis relies on the completeness of the real numbers, including those concerning the existence of limits of sequences, the behaviour of continuous functions, and the existence and uniqueness of solutions to differential equations.

Real analysis is the part of mathematical analysis, especially as taught in undergraduate and graduate courses, that develops calculus rigorously over the real numbers and Euclidean spaces. Introductory real analysis is sometimes called advanced calculus, and studies limits, continuity, compactness, differentiation, integration, and series. More advanced courses often include measure theory, Lebesgue integration, and function spaces. Real analysis is also known, especially in older books, as the theory of functions of a real variable, in contrast to the theory of complex variables.

Scope

Real numbers

The real numbers are the basic setting of real analysis, which begins with their construction. The real numbers are distinguished from the rational numbers by their completeness. Roughly speaking, the real numbers have no gaps. This completeness can be formalized in several equivalent ways, one of which is the least upper bound property. This states that if a non-empty set of real numbers is bounded above, meaning that all of its elements are less than some number (an upper bound), then there is a least upper bound, that is an upper bound that is smaller than all of the others.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".

Most of the theorems that are proved in real analysis rely on completeness in one way or another. Some examples where its relevance are most apparent are as follows. The convergence of bounded monotone sequences, that is sequences that are increasing (or decreasing), is essentially equivalent to the least upper bound property, stated in sequence form. Completeness is also reflected in the intermediate value theorem: the continuous image of an interval is again an interval, so continuous functions cannot create gaps.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".

Limits and convergence

The concept of a limit underlies many of the ideas of calculus, such as the derivative. It is therefore of fundamental importance in real analysis, which provides the rigorous justification for calculus. Limits describe how a sequence, function, or family of functions behave under a limiting process, such as letting an index tend to infinity, or a letting point become very large or approach another point. The formal language of limits defines continuity, differentiation, integration, infinite series, and various kinds of approximations and asymptotics.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".

A recurring problem in real analysis is not just whether a limit exists, but how well one object approximates another. A convergent sequence approximates its limit, or a differentiable function is approximated by a linear function coming from the derivative. But real analysis can provide not only tools to justify the existence of these limits and how to calculate them, but also quantitative estimates of how good the approximation is. One example of this is the Taylor remainder, which gives an effective and computable constant that determines how well the linear approximation (or higher-order Taylor polynomial) approximates a function on an interval.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".

For sequences of functions, real analysis distinguishes between different modes of convergence. A sequence of functions converges pointwise if it converges at every point, but, roughly speaking, the rate of convergence may vary from point to point. It converges uniformly if it converges at all points at a comparable rate. Uniform convergence can be understood intuitively in terms of the graphs of the functions in the sequence: it means that, for any given thin error band around the limiting function, all but finitely many functions in the sequence stay within the band. Pointwise convergence means that this is true for an error band around each point, but the finite set of functions which must be excluded varies from point to point.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".

For sequences of functions, pointwise convergence often fails to preserve operations on the limit function. For example, it is not generally true that the pointwise limit of a sequence of continuous functions is continuous, or that the integral of the functions in a sequence passes to the integral of the limit function. But the uniform limit of continuous functions is continuous, and one can exchange integration and uniform limits on suitable domains. Uniform convergence is therefore important for many applications of real analysis. Questions such as "when is differentiation under the integral allowed?" or "when can I integrate an infinite sum term-by-term?" are typical examples in which uniform convergence provides a simple answer.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".

Differentiation and regularity

Differentiation measures the local rate of change of a function. In one variable, the derivative gives the slope of the best linear approximation to a function near a point. This point of view extends to several variables, where differentiability is expressed in terms of approximation by a linear map.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".

In addition to proving rigorously the fundamental properties of the derivative from calculus, such as the chain rule, real analysis also establishes the main theorems about the derivative, such as the mean value theorem and some of its generalizations like the Cauchy mean value theorem. Roughly speaking, the mean value theorem relates the derivative of a function to its average rate of change over intervals. The mean value theorem is important because it leads to an effective way to estimate the error in certain approximations or averaging processes.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".

Real analysis studies not only existence of derivatives, but also degrees of regularity. A function may be continuous but nowhere differentiable, differentiable but not continuously differentiable, or smooth (having derivatives of all orders) but not analytic (equal to its Taylor series). One theorem that is sometimes shown at the elementary level is Darboux's theorem, that the derivative of a differentiable function satisfies the intermediate value property; weaker than continuity, but still a restricted class of functions. In more advanced analysis, regularity questions become central. The classical derivative turns out to be very inconvenient in many applications, such as to differential equations where many problems have a more natural variational formulation using integration rather than differentiation. The functions one must consider are often not differentiable, and so generalizations of differentiability, and even what constitutes a function, are considered. Regularity then involves questions about whether the generalized derivatives are honest derivatives, and how smooth solutions to equations are. This is a common theme in measure theory, partial differential equations, Sobolev spaces, and the calculus of variations.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".

Integration and measure

Integration in analysis gives rigorous meaning to averaging, accumulation, and area. The Riemann integral formalizes the integral using approximations by finite sums over intervals. The Riemann integral is put on a firm foundation in basic real analysis courses. The fundamental theorem of calculus, which relates integration and differentiation, is proved for the Riemann integral. The fundamental theorem underlies many exact computations in elementary integral calculus by reducing definite integrals to antiderivatives. The mean value theorem for integration states that the average value of a continuous function over an interval, defined by a Riemann integral, is equal to the value of the function at some point inside the interval. The mean value theorem is important in deriving analytic estimates in analytic applications; for example, one form of the Taylor remainder uses it.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".

The Lebesgue integral is introduced in more advanced real analysis. Instead of approximating the integral by finite sums using partitions of the domain, the Lebesgue integral partitions the range instead. While the idea is similar to the Riemann integral, it differs from the Riemann integral in that one must formulate the area of horizontal slabs that are not connected rectangles, but may be distributed over complicated sets in the domain. Thus formulating the Lebesgue integral leads naturally to measure theory, the fundamental question of which is how to measure subsets of the real line – that is, assign a notion of length – if the subsets are allowed to be very complicated.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".

One reason for the success of the Lebesgue integral in analysis is that it is more compatible with limiting processes. For example, under relatively mild hypotheses, the limit of the Lebesgue integrals of members of a sequence of functions that converges only pointwise is equal to the integral of the pointwise limit (dominated convergence theorem). Moreover, the Lebesgue integral admits a wider class of functions as integrable. The Riemann integral, although not widely used in mathematical analysis, still has its uses however: most numerical integration relies on the Riemann rather than Lebesgue approach to integration.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".

Measure theory treats length, area, volume, mass, and probability as instances of a general concept of measure. The theory allows one to identify functions that are equal except on a set of measure zero, and one says that such functions are equal almost everywhere. In probability theory, the relevant notion is almost surely. An example is tossing a fair coin an infinite sequence of times. It is possible that the coin could come up tails every time, but this event has zero probability: it can be essentially excluded from any probabilistic argument.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".

The monotone convergence theorem, Fatou's lemma, dominated convergence theorem, and Fubini's theorem are basic theorems about the Lebesgue integral.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters". Measure-theoretic real analysis also supplies the language of probability and of many function spaces. In probability theory, expected values are integrals with respect to probability measures. In the theory of Lp spaces, functions are studied according to integrability properties and are identified when they agree almost everywhere.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".

Series, function sequences, and representation

Script error: No such module "Labelled list hatnote". Infinite series are one of the basic limiting processes of real analysis. A numerical series is defined by the convergence of its partial sums, and questions about its absolute convergence, conditional convergence, and rearrangements illustrate the difference between finite and infinite additions. Power series and Taylor series connect infinite series with differentiablity and analyticity.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".

Sequences and series are used to approximate and represent functions. Uniform convergence gives conditions under which continuity, integrability, and sometimes differentiability, pass to the limit function. Power series can be used to represent some functions locally, while Fourier series represent periodic functions in terms of trigonometric series. These lead from elementary analysis to questions of approximation theory (how well a function is represented by a series or partial sum), harmonic analysis (what can be inferred about the regularity of the function from its representations), and the theory of function spaces (what classes of functions can be represented, and how well, in a certain way).Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".

Metric spaces and function spaces

Many of the theorems of real analysis can be formulated in metric spaces. Metric spaces generalize the real line and Euclidean spaces, in the sense that they have a notion of distance function defined on them that satisfies some natural rules. The basic theorems about limits and continuity carry through, essentially unchanged, in the more general setting, and so elementary treatments of the subject often describe this generalization.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters". The generalizations to metric spaces are important in many areas of mathematics. One example of a metric space is a surface in Euclidean space. The analytic properties of metrics like this are similar to those of Euclidean space, and differential geometry involves the study of metric spaces like surfaces and their generalizations to higher dimensions.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters". Other metric spaces in mathematics are very different from Euclidean spaces. In number theory for example, metric spaces are used to encode arithmetic information and invariants, a basic example of which is the p-adic numbers.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".

In real analysis proper, most of the examples of metric spaces other than Euclidean spaces are function spaces, that is spaces of functions that are defined by some property. In the basic examples considered in real analysis, these spaces of functions have a metric that comes from a norm. An example is the space of continuous functions on the unit interval, denoted C([0,1]). Given a real-valued function on the unit interval, the maximum absolute value of the function defines the norm. The metric is the maximum absolute difference between two functions. The notion of convergence that this metric defines is that of uniform convergence, so that the language of uniform convergence of continuous functions can be packaged into a single metric space. This is an example of a Banach space: the metric that this norm defines is complete, which is a consequence of a theorem that the uniform limit of continuous functions is continuous.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".

Another example of metric spaces appearing in real analysis, where the metric comes from a norm, is the space L1 of Lebesgue integrable functions, and the distance function is the integral of the absolute value of the difference of two functions. (Here functions are regarded as the same if they differ on a set of measure zero.) Thus, on a finite domain, the distance between two functions is essentially the average distance between their values. Spaces of square integrable measurable functions form a metric space, where the norm is induced by an inner product. This inner product is complete: the square integrable functions are a Hilbert space. The Lp spaces are generalizations of these. The metrics one gets from the Lp spaces are complete: they are Banach spaces, with the usual identification of measurable functions that are equal almost everywhere.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".

These function spaces, while all being infinite dimensional vector spaces, and share similar kinds of convergence theorems because they are all metric spaces, are different both globally as spaces, and locally in terms of the fine structure that functions must carry for membership. Functional analysis is concerned with these spaces and their generalizations, their global and convergence properties. Harmonic analysis is concerned with the fine properties needed for membership, and to estimate how operators determined by fine structure behave in theses spaces: a typical question here is to characterize when the derivative of a function is in a certain Lp space, for example..Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".

Completeness and compactness

Many applications of metric spaces to analysis rely on a metric space with extra properties that ensure that certain limits exist and belong to the space. One such property is whether a metric space is complete, and another is whether a metric space is compact.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".

Completeness of a space is formulated in terms of Cauchy sequences within it. A sequence in a space converges to a limit if all except finitely many of its members are arbitrarily close to the limit. Now, if a sequence converges, then all but finitely many of its members are close to the limit, and therefore must be close to one another. This property characterizes the Cauchy sequences in a metric space. Stated a little more carefully, a sequence is Cauchy if, for any error tolerance, all but finitely many members lie within that error tolerance of each other. A metric space is complete if every Cauchy sequence converges. The real numbers, for example, are complete in this sense.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".

One key theorem for complete metric spaces is the contraction mapping theorem. This theorem says that if a transformation T of a complete space decreases distances between points by at least a factor of C<1, so that dist⁡(Tx,Ty)≤Cdist⁡(x,y), then the transformation has a unique fixed point, that is a point of the space such that Tx=x. This theorem is particularly important in differential equations, because it supplies the basic existence and uniqueness theorem for solutions of ordinary differential equations – the Picard existence theorem – as well as a method for converging to it – Picard iteration. Furthermore, it provides estimates of the rate of convergence, provided one can determine the contractivity constant C.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".

Another notion in metric spaces is compactness. A metric space is compact if every sequence has a convergent subsequence. Compact metric spaces are also complete, but compactness is useful for guaranteeing the existence of limits without needing to study a separate Cauchy condition. In basic real analysis, the Bolzano–Weierstrass theorem shows that a subset of Euclidean space is compact if and only if it is closed and bounded.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".

A more general notion of compactness applies to general topological spaces that are not necessarily metric spaces, and it is based on the idea that compact spaces should generalized finite sets. Finite sets have the property that whenever they are covered by a family of subsets, so that every member of the set belongs to some member of the cover, it is possible to cover the finite set with a finite subfamily. For example, one can pick from each element of the finite set one member of the covering family that contains it. A metric space is compact in this sense if every covering by open sets has a finite subcover. An example of a compact set in this sense is a closed interval [a,b]: whenever this set is covered by open intervals, it is possible to cover it with finitely many of the intervals. This can be shown by a straightforward application of the least upper bound property. A compact set in the real line can be shown to be bounded, by considering an open cover by intervals (−n,n) and using a finite subcover to extract a bound. It can also be shown to be closed by a simple proof by contradiction. Another foundational result in real analysis is the Heine–Borel theorem, which states the converse to this: a subset of the real line (or Euclidean space, more generally) is compact in this sense if and only if it is closed and bounded.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".

Compactness plays an important role through its interaction with continuity. The extreme value theorem in calculus, for example, says that a continuous function on a closed interval [a,b] attains both a maximum and minimum value. In real analysis, this generalizes: any continuous real-valued function on a compact metric space attains both a maximum and minimum value. This result is important in optimization theory because it guarantees the existence of maxima and minima, often in very general situations where compactness is present but one is no longer working on subsets of the real line (or other Euclidean space).Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".

Another theorem in real analysis, relating to compactness and continuous functions, is the Arzelà–Ascoli theorem. It asserts that if a sequence of continuous functions on a compact metric space is uniformly bounded (all functions share a common upper and lower bound) and equicontinuous (all functions share a common modulus of continuity), then the sequence of functions has a uniformly convergent subsequence. This theorem has applications to differential equations, because it can be used to guarantee existence of solutions to differential equations where the stricter criteria of the Picard existence theorem do not hold. An example application is the Peano existence theorem. The Arzelà–Ascoli theorem is itself a kind of compactness assertion, characterizing compactness in the Banach space of functions on a compact metric space.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".

Important results

Important elementary results put the techniques of calculus on rigorous footing. They include the Bolzano–Weierstrass and Heine–Borel theorems, L'Hopital's rule, the mean value theorem, Taylor's theorem, the fundamental theorem of calculus, and the extreme value theorem.

Other results, also taught in an elementary course, are motivated by applications, chiefly to differential equations, multivariable calculus, and Fourier analysis which are part of the undergraduate mathematics curriculum. They are the Arzelà-Ascoli theorem, the Stone-Weierstrass theorem, the Banach fixed-point theorem, the inverse and implicit function theorems, and Stokes' theorem.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".

More advanced graduate-level courses also cover the basics of measure theory, the basic theorems of which are: Egorov's theorem, Lusin's theorem, Fatou's lemma, the monotone and dominated convergence theorems, and Fubini's theorem.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".

Other advanced results are sometimes also covered at the graduate level, because these are motivated by applications to other areas in the graduate curriculum. They include the Radon–Nikodym theorem, Lebesgue decomposition theorem, and Riesz representation theorem. Sometimes results such as the Lebesgue differentiation theorem are also included, as applications of harmonic analysis to questions of real analysis.

Generalizations and related areas of mathematics

Various ideas from real analysis can be generalized from the real line to broader or more abstract contexts. These generalizations link real analysis to other disciplines and subdisciplines. For instance, generalization of ideas like continuous functions and compactness from real analysis to metric spaces and topological spaces connects real analysis to the field of general topology, while generalization of finite-dimensional Euclidean spaces to infinite-dimensional analogs led to the concepts of Banach spaces and Hilbert spaces and, more generally to functional analysis. Georg Cantor's investigation of sets and sequence of real numbers, mappings between them, and the foundational issues of real analysis gave birth to naive set theory. The study of issues of convergence for sequences of functions eventually gave rise to Fourier analysis as a subdiscipline of mathematical analysis. Investigation of the consequences of generalizing differentiability from functions of a real variable to ones of a complex variable gave rise to the concept of holomorphic functions and the inception of complex analysis as another distinct subdiscipline of analysis. On the other hand, the generalization of integration from the Riemann sense to that of Lebesgue led to the formulation of the concept of abstract measure spaces, a fundamental concept in measure theory. Finally, the generalization of integration from the real line to curves and surfaces in higher dimensional space brought about the study of vector calculus, whose further generalization and formalization played an important role in the evolution of the concepts of differential forms and smooth (differentiable) manifolds in differential geometry and other closely related areas of geometry and topology.Script error: No such module "Unsubst".

Distributions

Script error: No such module "Labelled list hatnote". Distributions (or generalized functions) are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative.

Relation to complex analysis

Real analysis is an area of analysis that studies concepts such as sequences and their limits, continuity, differentiation, integration and sequences of functions. By definition, real analysis focuses on the real numbers, often including positive and negative infinity to form the extended real line. Real analysis is closely related to complex analysis, which studies broadly the same properties of complex numbers. In complex analysis, it is natural to define differentiation via holomorphic functions, which have a number of useful properties, such as repeated differentiability, expressibility as power series, and satisfying the Cauchy integral formula.

In real analysis, it is usually more natural to consider differentiable, smooth, or harmonic functions, which are more widely applicable, but may lack some more powerful properties of holomorphic functions. However, results such as the fundamental theorem of algebra are simpler when expressed in terms of complex numbers.

Techniques from the theory of analytic functions of a complex variable are often used in real analysis – such as evaluation of real integrals by residue calculus.

See also

References

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Sources

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Bibliography

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External links

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