Geodesic
Singular points of order $k$ of Clarke regular and arbitrary functions
Commentationes Mathematicae Universitatis Carolinae, Tome 53 (2012) no. 1, pp. 51-63

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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).
Classification : 26B25, 49J52
Keywords: Clarke regular functions; singularities; Hadamard derivative 
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     title = {Singular points of order $k$ of {Clarke} regular and arbitrary functions},
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Zajíček, Luděk. Singular points of order $k$ of Clarke regular and arbitrary functions. Commentationes Mathematicae Universitatis Carolinae, Tome 53 (2012) no. 1, pp. 51-63. http://geodesic.mathdoc.fr/item/CMUC_2012__53_1_a3/