On the subdifferentiability of convex functions

Abstract

(Thus the subgradients of f correspond to the nonvertical supporting hyperplanes to the convex set consisting of all the points of E (DR lying above the graph of f.) The set of subgradients of f at x is denoted by of(x). If of(x) is not empty, f is said to be subdifferenticable at x. Iff actually had a gradient x* = Vf(x) at x in the sense of Gateaux (or Frechet), one would in particular have af(x) = { Vf(x) } (see Moreau [5, p. 20]). It is immediate from (1.2) that of(x) is a weak* closed convex set in E* for each xCE, and that the effective domain