# Regulated function

In mathematics, a regulated function, or ruled function, is a certain kind of well-behaved function of a single real variable. Regulated functions arise as a class of 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:
* for every t in the interval [0, T], both the left and right limits f(t−) and f(t+) exist in X (apart from, obviously, f(0−) and f(T+));
* there exists a sequence of step functions φ$_n$ : [0, T] → X 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
       $\|f-\varphi _{\delta }\|_{\infty }=\sup _{t\in [0,T]}\|f(t)-\varphi _{\delta }(t)\|_{X}<\delta ;$
* f lies in the 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 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 K as the space X; typically, K will be the real or complex numbers. 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, then so is Reg([0, T]; X).
* The supremum norm is a 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, 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 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 linear subspace of Reg([0, T]; X):
       $\mathrm {Reg} ([0,T];X)={\overline {\mathrm {BV} ([0,T];X)}}{\mbox{ w.r.t. }}\|\cdot \|_{\infty }.$
* If X is a Banach space, then a function f : [0, T] → X is regulated if and only if it is of bounded φ-variation for some φ:
       $\mathrm {Reg} ([0,T];X)=\bigcup _{\varphi }\mathrm {BV} _{\varphi }([0,T];X).$
* If X is a separable Hilbert space, then Reg([0, T]; X) satisfies a compactness theorem known as the Fraňková–Helly selection theorem.
* The set of 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 $\epsilon >0$, the set of points at which the right and left limits differ by more than $\epsilon$ 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 $F_{\sigma }$ is either meager or else has nonempty interior. This is not always equivalent with countability.
* 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 space that is dual to Reg([0, T]; R);
** agrees with the Riemann integral.