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 $\operatorname {int} D \ne \emptyset$, then, for each Gateaux differentiability point x $( \in \operatorname {int} D)$ of f, there is a closed convex set $C \subset \operatorname {int} D$ with the nonsupport points set $N(C) \ne \emptyset$ and with $x \in N(C)$ such that ${f_C}$ (the restriction of f on C) is Fréchet differentiable at x.