Some properties of semismooth and regular functions in nonsmooth analysis

Abstract

Given a real valued function f, defined on a locally convex topological space X, locally Lipschitzian, and Gateaux-differentiable on a dense subset D in X, we have studied under what hypotheses Charke's generalized gradient can be written as

$$\partial f(x) = \overline {co} {\text{ }}\{ w^* \mathop {\lim }\limits_{y \to x} \nabla f(y)/ y \in D\} {\text{ }},$$

It is shown that this formula is valid in particular when f is regular or semismooth. By using this characterization, some properties known to hold true in finite dimension are generalized and other new properties are established. In particular, a characterization of semismooth functions is given in terms of the continuity of the directional derivative. Finally, characterizations for the directional derivative and generalized gradient of marginal functions are obtained. In particular, Mifflin's result stating that lower-C¹ functions are semismooth is generalized.

Partially supported by Univ. de La Serena, under grant №130.2.07 and Fondo Nacional de Ciencias, under grant № 01273.