MATHEMATICA BOHEMICA, Vol. 132, No. 1, pp. 75-85 (2007)

On sets of non-differentiability of
Lipschitz and convex functions

Ludek Zajicek

Ludek Zajicek, Charles University, Faculty of Mathematics and Physics, Sokolovska 83, 186 75 Praha 8, Czech Republic, e-mail:

Abstract: We observe that each set from the system $\widetilde {\Cal A}$ (or even $\widetilde {\C }$) is $\Gamma $-null; consequently, the version of Rademacher's theorem (on Gateaux differentiability of Lipschitz functions on separable Banach spaces) proved by D. Preiss and the author is stronger than that proved by D. Preiss and J. Lindenstrauss. Further, we show that the set of non-differentiability points of a convex function on $\R ^n$ is $\sigma $-strongly lower porous. A discussion concerning sets of Frechet non-differentiability points of continuous convex functions on a separable Hilbert space is also presented.

Keywords: Lipschitz function, convex function, Gateaux differentiability, Frechet differentiability, $\Gamma $-null sets, ball small sets, $\delta $-convex surfaces, strong porosity

Classification (MSC2000): 46G05, 26B05

Full text of the article:

[Previous Article] [Next Article] [Contents of this Number] [Journals Homepage]
© 2007–2010 FIZ Karlsruhe / Zentralblatt MATH for the EMIS Electronic Edition