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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
DOI: https://doi.org/10.1090/S0002-9939-1980-0572313-1
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.


References [Enhancements On Off] (What's this?)

  • [1] C. L. Belna, M. J. Evans and P. D. Humke, Symmetric and ordinary differentiation, Proc. Amer. Math. Soc. 72 (1978), 261-267. MR 507319 (80d:26006)
  • [2] E. P. Dolženko, Boundary properties of arbitrary functions, Math. USSR-Izv. 1 (1967), 1-12.
  • [3] M. Kulbacka, Sur l'ensemble des points de l'asymetrie approximative, Acta Sci. Math. (Szeged) 21 (1960), 90-95. MR 0117307 (22:8088)
  • [4] L. Mišík, Über approximative derivierte Zahlen monotoner Funktionen, Czechoslovak Math. J. 26 (101) (1976), 579-583. MR 0432834 (55:5814)
  • [5] C. J. Neugebauer, A theorem on derivates. Acta Sci. Math. (Szeged) 23 (1962), 79-81. MR 0140624 (25:4041)
  • [6] L. Zajíček, On cluster sets of arbitrary functions, Fund. Math. 83 (1974), 197-217. MR 0338294 (49:3060)
  • [7] -, Sets of $ \sigma $-porosity and sets of $ \sigma $-porosity (q), Časopis Pěst. Mat. 101 (1976), 350-359. MR 0457731 (56:15935)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1980-0572313-1
Keywords: Derivates, bounded variation, monotone functions, $ \sigma $-porosity
Article copyright: © Copyright 1980 American Mathematical Society

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