Characterization of nonsmooth functions through their generalized gradients

Abstract

Given a real-valued Lipschitz function, we explore properties of its generalized gradient in Clarke's sense that corresponds to f being quasi-convex, to f being d.c, (difference of convex functions), to f being a pointwise supremum of functions that are ktimes continuously differentiable. In other words, knowing that ∂f is the generalized gradient multifunction of a function f what kind of properties of ∂f could serve to characterize f?. The classes of nonsmooth function involed in this paper are: quasi-function, d.e functions, lower-C ^k functions, semi smooth functions and quasidifferentiable functions.