## Singular points of order $$k$$ of Clarke regular and arbitrary functions.(English)Zbl 1249.49021

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 [J. Convex Anal. 17, No. 3–4, 1019–1032 (2010; Zbl 1208.46043)], 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).

### MSC:

 49J52 Nonsmooth analysis 26B25 Convexity of real functions of several variables, generalizations

### Keywords:

Clarke regular function; singularity; Hadamard derivative

Zbl 1208.46043
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