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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Of regulated and steplike functions


Author: Gadi Moran
Journal: Trans. Amer. Math. Soc. 231 (1977), 249-257
MSC: Primary 26A30
MathSciNet review: 0499028
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Abstract: 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 I. 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 I vanishing at 0, at the expense of a rearrangement of s.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1977-0499028-7
PII: S 0002-9947(1977)0499028-7
Keywords: Regulated functions, step functions, rearrangements of series of functions
Article copyright: © Copyright 1977 American Mathematical Society