A note on the differentiability of convex functions
Cong Xin Wu, Li Xin Cheng · Proceedings of the American Mathematical Society · 1994
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 ≠ ∅ \operatorname {int} D e \emptyset , then, for each Gateaux differentiability point x ( ∈ int D ) ( \in \operatorname {int} D) of f, there is a closed convex set C ⊂ int D C \subset \operatorname {int} D with the nonsupport points set N ( C ) ≠ ∅ N(C) e \emptyset and with x ∈ N ( C ) x \in N(C) such that f C {f_C} (the restriction of f on C) is Fréchet differentiable at x.