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Title: Singular points of order $k$ of Clarke regular and arbitrary functions (English)
Author: Zajíček, Luděk
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 53
Issue: 1
Year: 2012
Pages: 51-63
Summary lang: English
Category: math
Summary: Let $X$ be a separable Banach space and $f$ a locally Lipschitz real function on $X$. For $k\in \mathbb N$, let $\Sigma_k(f)$ be the set of points $x\in X$, at which the Clarke subdifferential $\partial^Cf(x)$ is at least $k$-dimensional. It is well-known that if $f$ is convex or semiconvex (semiconcave), then $\Sigma_k(f)$ can be covered by countably many Lipschitz surfaces of codimension $k$. We show that this result holds even for each Clarke regular function (and so also for each approximately convex function). Motivated by a resent result of A.D. Ioffe, we prove also two results on arbitrary functions, which work with Hadamard directional derivatives and can be considered as generalizations of our theorem on $\Sigma_k(f)$ of Clarke regular functions (since each of them easily implies this theorem). (English)
Keyword: Clarke regular functions
Keyword: singularities
Keyword: Hadamard derivative
MSC: 26B25
MSC: 49J52
idZBL: Zbl 1249.49021
idMR: MR2880910
Date available: 2012-02-07T10:23:17Z
Last updated: 2014-04-07
Stable URL:
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