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In [[mathematics]], a '''regulated function''', or '''ruled function''', is a certain kind of well-behaved [[function (mathematics)|function]] of a single [[real number|real]] variable. Regulated functions arise as a class of [[integration (mathematics)|integrable functions]], and have several equivalent characterisations. Regulated functions were introduced by [[Nicolas Bourbaki]] in 1949, in their book "Livre IV: Fonctions d'une variable réelle".

==Definition==

Let ''X'' be a [[Banach space]] with norm || - ||''X''. A function ''f'' : [0, ''T''] → ''X'' is said to be a '''regulated function''' if one (and hence both) of the following two equivalent conditions holds true:<ref>{{harvnb|Dieudonné|1969|loc=§7.6}}</ref>

* for every ''t'' in the [[interval (mathematics)|interval]] [0, ''T''], both the [[limit of a function|left and right limits]] ''f''(''t''−) and ''f''(''t''+) exist in ''X'' (apart from, obviously, ''f''(0−) and ''f''(''T''+));
* there exists a [[sequence (mathematics)|sequence]] of [[step function]]s ''φ''''n'' : [0, ''T''] → ''X'' [[uniform convergence|converging uniformly]] to ''f'' (i.e. with respect to the [[supremum norm]] || - ||∞).

It requires a little work to show that these two conditions are equivalent. However, it is relatively easy to see that the second condition may be re-stated in the following equivalent ways:

* for every ''δ'' > 0, there is some step function ''φ''''δ'' : [0, ''T''] → ''X'' such that

::<math>\| f - \varphi_\delta \|_\infty = \sup_{t \in [0, T]} \| f(t) - \varphi_\delta (t) \|_X < \delta;</math>

* ''f'' lies in the [[closed set|closure]] of the space Step([0, ''T'']; ''X'') of all step functions from [0, ''T''] into ''X'' (taking closure with respect to the supremum norm in the space B([0, ''T'']; ''X'') of all bounded functions from [0, ''T''] into ''X'').

==Properties of regulated functions==

Let Reg([0, ''T'']; ''X'') denote the [[Set (mathematics)|set]] of all regulated functions ''f'' : [0, ''T''] → ''X''.

* Sums and scalar multiples of regulated functions are again regulated functions. In other words, Reg([0, ''T'']; ''X'') is a [[vector space]] over the same [[field (mathematics)|field]] '''K''' as the space ''X''; typically, '''K''' will be the [[real number|real]] or [[complex number]]s. If ''X'' is equipped with an operation of multiplication, then products of regulated functions are again regulated functions. In other words, if ''X'' is a '''K'''-[[Algebra over a field|algebra]], then so is Reg([0, ''T'']; ''X'').
* The supremum norm is a [[norm (mathematics)|norm]] on Reg([0, ''T'']; ''X''), and Reg([0, ''T'']; ''X'') is a [[topological vector space]] with respect to the topology induced by the supremum norm.
* As noted above, Reg([0, ''T'']; ''X'') is the closure in B([0, ''T'']; ''X'') of Step([0, ''T'']; ''X'') with respect to the supremum norm.
* If ''X'' is a [[Banach space]], then Reg([0, ''T'']; ''X'') is also a Banach space with respect to the supremum norm.
* Reg([0, ''T'']; '''R''') forms an infinite-dimensional real [[Banach algebra]]: finite linear combinations and products of regulated functions are again regulated functions.
* Since a [[continuous function]] defined on a [[compact space]] (such as [0, ''T'']) is automatically [[uniformly continuous function|uniformly continuous]], every continuous function ''f'' : [0, ''T''] → ''X'' is also regulated. In fact, with respect to the supremum norm, the space ''C''0([0, ''T'']; ''X'') of continuous functions is a [[closed set|closed]] [[linear subspace]] of Reg([0, ''T'']; ''X'').
* If ''X'' is a [[Banach space]], then the space BV([0, ''T'']; ''X'') of functions of [[bounded variation]] forms a [[dense set|dense]] linear subspace of Reg([0, ''T'']; ''X''):

::<math>\mathrm{Reg}([0, T]; X) = \overline{\mathrm{BV} ([0, T]; X)} \mbox{ w.r.t. } \| \cdot \|_{\infty}.</math>

* If ''X'' is a Banach space, then a function ''f'' : [0, ''T''] → ''X'' is regulated [[if and only if]] it is of [[Bounded variation#Weighted BV functions|bounded ''φ''-variation]] for some ''φ'':

::<math>\mathrm{Reg}([0, T]; X) = \bigcup_{\varphi} \mathrm{BV}_{\varphi} ([0, T]; X).</math>

* If ''X'' is a [[separable space|separable]] [[Hilbert space]], then Reg([0, ''T'']; ''X'') satisfies a compactness theorem known as the [[Fraňková–Helly selection theorem]].
* The set of [[Discontinuity (mathematics)|discontinuities]] of a regulated function of [[bounded variation]] BV is [[countable]] for such functions have only jump-type of discontinuities. To see this it is sufficient to note that given <math> \epsilon > 0 </math>, the set of points at which the right and left limits differ by more than <math> \epsilon</math> is finite. In particular, the discontinuity set has [[measure zero]], from which it follows that a regulated function has a well-defined [[Riemann integral]].
* Remark: By the Baire Category theorem the set of points of discontinuity of such function <math>F_\sigma</math> is either meager or else has nonempty interior. This is not always equivalent with countability.<ref>[https://math.stackexchange.com/q/84870 Stackexchange discussion]</ref>

* The integral, as defined on step functions in the obvious way, extends naturally to Reg([0, ''T'']; ''X'') by defining the integral of a regulated function to be the limit of the integrals of any sequence of step functions converging uniformly to it. This extension is [[well-defined]] and satisfies all of the usual properties of an integral. In particular, the [[regulated integral]]
** is a [[bounded linear function]] from Reg([0, ''T'']; ''X'') to ''X''; hence, in the case ''X'' = '''R''', the integral is an element of the [[continuous dual space|space that is dual]] to Reg([0, ''T'']; '''R''');
** agrees with the [[Riemann integral]].

==References==
{{Reflist}}

* {{citation
| first = Georg
| last = Aumann
| authorlink = Georg Aumann
| title = Reelle Funktionen
| language = German
| series = Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Bd LXVIII
| publisher = Springer-Verlag
| location = Berlin
| year = 1954
| pages = viii+416
}} {{MathSciNet| id = 0061652}}
* {{citation
| first = Jean
| last = Dieudonné
| authorlink = Jean Dieudonné
| title = Foundations of Modern Analysis
| publisher = Academic Press
| year = 1969
| pages = xviii+387
}} {{MathSciNet | id = 0349288}}
* {{citation
| last = Fraňková
| first = Dana
| title = Regulated functions
| journal = Math. Bohem.
| volume = 116
| year = 1991
| pages = 20–59
| issn = 0862-7959
| issue = 1
}} {{MathSciNet | id = 1100424}}
* {{citation
| last = Gordon
| first = Russell A.
| title = The Integrals of Lebesgue, Denjoy, Perron, and Henstock
| series = [[Graduate Studies in Mathematics]], 4
| publisher = American Mathematical Society
| location = Providence, RI
| year = 1994
| pages = [https://archive.org/details/integralsoflebes0004gord/page/ xii+395]
| isbn = 0-8218-3805-9
| url = https://archive.org/details/integralsoflebes0004gord/page/
}} {{MathSciNet | id = 1288751}}
* {{citation
| last = Lang
| first = Serge
| authorlink = Serge Lang
| title = Differential Manifolds
| edition = Second
| publisher = Springer-Verlag
| location = New York
| year = 1985
| pages = ix+230
| isbn = 0-387-96113-5
}} {{MathSciNet | id = 772023}}

==External links==

*{{cite web |title=How to show that a set of discontinuous points of an increasing function is at most countable |date=November 23, 2011 |work=[[Stack Exchange]] |url=https://math.stackexchange.com/q/84870 }}
*{{cite web |title=Bounded variation functions have jump-type discontinuities |date=November 28, 2013 |work=Stack Exchange |url=https://math.stackexchange.com/q/584735 }}
*{{cite web |title=How discontinuous can a derivative be? |date=February 22, 2012 |work=Stack Exchange |url=https://math.stackexchange.com/q/112067 }}

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

Regulated function - Revision history

imported>RL0919: more consistent citation format