| 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:
|
http://hdl.handle.net/10338.dmlcz/141825 |
| . |
| Reference:
|
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| Reference:
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| Reference:
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| Reference:
|
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| . |
