Luděk Zajíček
Singular points of order $k$ of Clarke regular and arbitrary functions

Comment.Math.Univ.Carolin. 53,1 (2012) 51-63.

Abstract: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).

Keywords: Clarke regular functions, singularities, Hadamard derivative
AMS Subject Classification: 49J52 26B25