Given a real-valued Lipschitz function, the authors 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 k-times 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 functions involved in this paper are: quasi-convex functions, d.c. functions, lower-Ck functions, semi smooth functions and quasidifferentiable functions.
