The equality of unilateral derivates

Evans, M. J.; Humke, P. D.

doi:10.1090/S0002-9939-1980-0572313-1

The equality of unilateral derivates
HTML articles powered by AMS MathViewer

by M. J. Evans and P. D. Humke

Proc. Amer. Math. Soc. 79 (1980), 609-613

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

PDF | Request permission

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

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, DOI 10.1090/S0002-9939-1978-0507319-2

Boundary properties of arbitrary functions

1

Marie Kulbacka, Sur l’ensemble des points de l’asymétrie approximative, Acta Sci. Math. (Szeged) 21 (1960), 90–95 (French). MR 117307
Ladislav Mišík, Über approximative derivierte Zahlen monotoner Funktionen, Czechoslovak Math. J. 26(101) (1976), no. 4, 579–583. MR 432834
C. J. Neugebauer, A theorem on derivates, Acta Sci. Math. (Szeged) 23 (1962), 79–81. MR 140624
L. Zajíček, On cluster sets of arbitrary functions, Fund. Math. 83 (1973/74), no. 3, 197–217. MR 338294, DOI 10.4064/fm-83-3-197-217
Luděk Zajíček, Sets of $\sigma$-porosity and sets of $\sigma$-porosity $(q)$, Časopis Pěst. Mat. 101 (1976), no. 4, 350–359 (English, with Russian summary). MR 0457731