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): 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: 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 , the set of points at which the right and left limits differ by more than 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 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. References External links Real analysis Types of functions