A NOTE ON THE DIFFERENTIABILITY OF CONVEX-FUNCTIONS

Every real-valued convex and locally Lipschitzian function f defined on a nonempty closed convex set D of a Banach space E is the local restriction of a convex Lipschitzian function defined on E. Moreover, if E is separable and int D not-equal empty set, then, for each Gateaux differentiability point x (is-an-element-of int D) of f, there is a closed convex set C subset-of int D with the nonsupport points set N(C) not-equal empty set and with x is-an-element-of N(C) such that f(C) (the restriction of f on C) is Frechet differentiable at x.