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In [[mathematics]], '''Baire functions''' are [[function (mathematics)|function]]s obtained from [[continuous functions]] by transfinite iteration of the operation of forming pointwise limits of sequences of functions. They were introduced by [[René-Louis Baire]] in 1899. A [[Baire set]] is a set whose [[indicator function|characteristic function]] is a Baire function.

==Classification of Baire functions==
Baire functions of class α, for any countable [[ordinal number]] α, form a [[vector space]] of [[Real number|real]]-valued functions defined on a [[topological space]], as follows.<ref>{{cite journal |last=Jech |first=Thomas |date=November 1981 |title=The Brave New World of Determinacy |journal=Bulletin of the American Mathematical Society |volume=5 |number=3 |pages=339–349 |doi=10.1090/S0273-0979-1981-14952-1 |url=https://projecteuclid.org/journalArticle/Download?urlid=bams%2F1183548432|doi-access=free }}</ref>

*The Baire class 0 functions are the [[Continuous function (topology)|continuous function]]s.
*The Baire class 1 functions are those functions which are the [[Pointwise convergence|pointwise limit]] of a [[sequence]] of Baire class 0 functions.
*In general, the Baire class α functions are all functions which are the pointwise limit of a sequence of functions of Baire class less than α.

Some authors define the classes slightly differently, by removing all functions of class less than α from the functions of class α. This means that each Baire function has a well defined class, but the functions of given class no longer form a vector space.

[[Henri Lebesgue]] proved that (for functions on the [[unit interval]]) each Baire class of a countable ordinal number contains functions not in any smaller class, and that there exist functions which are not in any Baire class.

==Baire class 1==
Examples:
*The [[derivative]] of any [[differentiable function]] is of class 1. An example of a differentiable function whose derivative is not continuous (at ''x'' = 0) is the function equal to <math>x^2 \sin(1/x)</math> when ''x'' ≠ 0, and 0 when ''x'' = 0. An infinite sum of similar functions (scaled and displaced by [[rational number]]s) can even give a differentiable function whose derivative is discontinuous on a [[dense set]]. However, it necessarily has points of continuity, which follows easily from The Baire Characterisation Theorem (below; take ''K'' = ''X'' = '''R''').
*The characteristic function of the set of [[integer]]s, which equals 1 if ''x'' is an integer and 0 otherwise. (An infinite number of large discontinuities.)
*[[Thomae's function]], which is 0 for [[Irrational number|irrational]] ''x'' and 1/''q'' for a rational number ''p''/''q'' (in reduced form). (A dense set of discontinuities, namely the set of rational numbers.)
*The characteristic function of the [[Cantor set]], which equals 1 if ''x'' is in the Cantor set and 0 otherwise. This function is 0 for an [[uncountable set]] of ''x'' values, and 1 for an uncountable set. It is discontinuous wherever it equals 1 and continuous wherever it equals 0. It is approximated by the continuous functions <math>g_n(x) = \max(0,{1-nd(x,C)})</math>, where <math>d(x,C)</math> is the distance of x from the nearest point in the Cantor set.

The Baire Characterisation Theorem states that a real valued function ''f'' defined on a [[Banach space]] ''X'' is a Baire-1 function [[if and only if]] for every [[Empty set|non-empty]] [[Metric_space#Open_and_closed_sets,_topology_and_convergence|closed]] subset ''K'' of ''X'', the [[Restriction (mathematics)|restriction]] of ''f'' to ''K'' has a point of continuity relative to the [[Topological space|topology]] of ''K''.

By another theorem of Baire, for every Baire-1 function the points of continuity are a [[comeager]] [[Gδ set|''G''<sub>δ</sub> set]] {{harv|Kechris|1995|loc=Theorem (24.14)}}.

==Baire class 2==
An example of a Baire class 2 function on the interval [0,1] that is not of class 1 is the characteristic function of the rational numbers, <math>\chi_\mathbb{Q}</math>, also known as the [[Dirichlet function]] which is [[nowhere continuous function|discontinuous everywhere]].
{{Math proof|title=Proof|drop=hidden|proof=We present two proofs.
# This can be seen by noting that for any finite collection of rationals, the characteristic function for this set is Baire 1: namely the function <math>g_n(x) = \max(0,{1-nd(x,K)})</math> converges identically to the characteristic function of <math>K</math>, where <math>K</math> is the finite collection of rationals. Since the rationals are countable, we can look at the pointwise limit of these things over <math>K_n = \{r_1,r_2,\dots,r_n\}</math>, where <math>r_n</math> is an enumeration of the rationals. It is not Baire-1 by the theorem mentioned above: the set of discontinuities is the entire interval (certainly, the set of points of continuity is not comeager).
# The Dirichlet function can be constructed as the double pointwise limit of a sequence of continuous functions, as follows:
:::<math>\forall x \in \mathbb R,\quad\chi_{\mathbb Q}(x) = \lim_{k\to\infty}\left(\lim_{j\to\infty}\left(\cos(k!\pi x)\right)^{2j}\right)</math>
:: for integer ''j'' and ''k''.}}

==See also==
*[[Baire set]]
*[[Nowhere continuous function]]

==References==
===Inline===
{{Reflist}}

===General===
*{{cite thesis |type=Ph.D. |last=Baire |first=René-Louis |year=1899 |title=Sur les fonctions de variables réelles |language=French |publisher=École Normale Supérieure |url=https://archive.org/details/surlesfonctions00bairgoog}}
*{{citation |last=Baire |first=René-Louis |year=1905 |title=Leçons sur les fonctions discontinues, professées au collège de France |language=French |publisher=Gauthier-Villars}}
*{{citation |last=Kechris |first=Alexander S. |author-link=Alexander S. Kechris |year=1995 |title=Classical Descriptive Set Theory |series=Graduate Texts in Mathematics |publisher=Springer-Verlag |isbn=978-1-4612-8692-9 |volume=156}}

==External links==
*[https://encyclopediaofmath.org/wiki/Baire_classes Springer Encyclopaedia of Mathematics article on Baire classes]

[[Category:General topology]]
[[Category:Real analysis]]
[[Category:Types of functions]]

Baire function - Revision history

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