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

Zajíček, Luděk

Commentat. Math. Univ. Carol. 53, No. 1, 51-63 (2012).

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

Citations:

Zbl 1208.46043