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 || - ||∞). 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 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