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 | References | Similar Articles | Additional Information

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 . It is also shown that if *f* is monotone, then *E* is -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 -porous set of points, a result first proved by L. Zajíček.

**[1]**C. L. Belna, M. J. Evans, and P. D. Humke,*Symmetric and ordinary differentiation*, Proc. Amer. Math. Soc.**72**(1978), no. 2, 261–267. MR**507319**, https://doi.org/10.1090/S0002-9939-1978-0507319-2**[2]**E. P. Dolženko,*Boundary properties of arbitrary functions*, Math. USSR-Izv.**1**(1967), 1-12.**[3]**Marie Kulbacka,*Sur l’ensemble des points de l’asymétrie approximative*, Acta Sci. Math. Szeged**21**(1960), 90–95 (French). MR**0117307****[4]**Ladislav Mišík,*Über approximative derivierte Zahlen monotoner Funktionen*, Czechoslovak Math. J.**26(101)**(1976), no. 4, 579–583. MR**0432834****[5]**C. J. Neugebauer,*A theorem on derivates*, Acta Sci. Math. (Szeged)**23**(1962), 79–81. MR**0140624****[6]**L. Zajíček,*On cluster sets of arbitrary functions*, Fund. Math.**83**(1973/74), no. 3, 197–217. MR**0338294**, https://doi.org/10.4064/fm-83-3-197-217**[7]**Luděk Zajíček,*Sets of 𝜎-porosity and sets of 𝜎-porosity (𝑞)*, Časopis Pěst. Mat.**101**(1976), no. 4, 350–359 (English, with Loose Russian summary). MR**0457731**

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Additional Information

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

Keywords:
Derivates,
bounded variation,
monotone functions,
-porosity

Article copyright:
© Copyright 1980
American Mathematical Society