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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The equality of unilateral derivates


Authors: M. J. Evans and P. D. Humke
Journal: Proc. Amer. Math. Soc. 79 (1980), 609-613
MSC: Primary 26A24; Secondary 26A45
MathSciNet review: 572313
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Abstract: C. J. Neugebauer has shown that if f is a continuous function of bounded variation defined on the real line, then the set E where the upper right derivate differs from the upper left derivate is of measure zero and first category. Here it is shown that this result is best possible; that is, given any measure zero first category set K, there is a continuous function of bounded variation for which $ K \subseteq E$. It is also shown that if f is monotone, then E is $ \sigma $-porous. This result can be used to provide an elementary proof of the fact that for an arbitrary function f the left and right essential cluster sets are identical except at a $ \sigma $-porous set of points, a result first proved by L. Zajíček.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1980-0572313-1
PII: S 0002-9939(1980)0572313-1
Keywords: Derivates, bounded variation, monotone functions, $ \sigma $-porosity
Article copyright: © Copyright 1980 American Mathematical Society