Weakened subdifferentials and Frechet differentiability of real functions

Abstract

Let X be a real Banach space and f : X ! R [ {+1}. It is well known that the Clarke subdifferential @ f(x) of the function f at x 2 int dom f is a singleton if and only if f is strongly differentiable (then @ f(x) = {Dsf(x)}, where Dsf(x) is the strong subdifferential of f at x). Simple examples show that there exist Fr´echet differentiable at x functions f, for which @ f(x) is not a singleton. In such a sense the Clarke subdifferential is not an exact generalization of the differential of a differentiable function. In the present paper we propose a new subdifferential @w f(x), called the weakened subdifferential of f at x, which preserves the nice calculus rules of the Clarke subdifferential, and for X finite dimensional, is a singleton @w f(x) = {} if and only if f is Fr´echet differentiable at x, and then = DF f(x).generalized subdifferentials.