2005): Clarke critical values of subanalytic Lipschitz continuous functions

Abstract. We prove that any subanalytic locally Lipschitz function has the Sard property. Such functions are typically nonsmooth and their lack of regularity necessitates the choice of some generalized no-tion of gradient and of critical point. In our framework these notions are defined in terms of the Clarke and of the convex-stable subdiffer-entials. The main result of this note asserts that for any subanalytic locally Lipschitz function the set of its Clarke critical values is lo-cally finite. The proof relies on Pawlucki’s extension of the Puiseux lemma. In the last section we give an example of a continuous subana-lytic function which is not constant on a segment of “broadly critical” points, that is, points for which we can find arbitrarily short convex combinations of gradients at nearby points. 1