Of regulated and steplike functions

Let C denote the class of regulated real-valued functions on the unit interval vanishing at the origin, whose positive and negative jumps sum to infinity in every nontrivial subinterval of f. Goffman [2] showed that every f in C is (essentially) a sum g + s where g is continuous and s is steplike. In this sense, a function in C is like a function of bounded variation, that has a unique such g and s. The import of this paper is that for 𝑓 in C the representation 𝑓 = g + s is not only not unique, but by far the opposite holds: g can be chosen to be any continuous function on f vanishing at 0, at the expense of a rearrangement of s.