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 φn : [0, T] → X converging uniformly to f (i.e. with respect to the supremum norm || - ||∞). For this equivalence to hold, the definition of "step function" must be the one that allows degenerate intervals (i.e., singletons). 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 φδ : [0, T] → X such that
‖ f − φ δ ‖ ∞ = sup t ∈ [ 0 , T ] ‖ f ( t ) − φ δ ( t ) ‖ X < δ ; {\displaystyle \|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 C0([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):
R e g ( [ 0 , T ] ; X ) = B V ( [ 0 , T ] ; X ) ¯ w.r.t. ‖ ⋅ ‖ ∞ . {\displaystyle \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:
R e g ( [ 0 , T ] ; X ) = ⋃ φ B V φ ( [ 0 , T ] ; X ) . {\displaystyle \mathrm {Reg} ([0,T];X)=\bigcup _{\varphi }\mathrm {BV} _{\varphi }([0,T];X).}
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