Of regulated and steplike functions

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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.

Original languageEnglish
Pages (from-to)249-257
Number of pages9
JournalTransactions of the American Mathematical Society
Issue number1
StatePublished - 1977


  • Rearrangements of series of functions
  • Regulated functions
  • Step functions

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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