Regulated functions and the regulated integral

Let I = [a, b] and let X be a normed space. A function f : I → X is said to be regulated if for all t ∈ [a, b) the limit lims→t+ f(s) exists and for all t ∈ (a, b] the limit lims→t− f(s) exists. We denote these limits respectively by f(t ) and f(t−). We define R(I,X) to be the set of regulated functions I → X. It is apparent that R(I,X) is a vector space. One checks that a regulated function is bounded, and that R(I,X) is a normed space with the norm ‖f‖∞ = supt∈[a,b] ‖f(t)‖. Theorem 1. If I is a compact interval in R and X is a normed algebra, then R(I,X) is a normed algebra.