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This paper presents a property of geometric and topological nature of Gateaux differentiability points and Frechet differentiability points of almost CL-spaces. More precisely, if we denote by M a maximal convex set of the unit sphere of a CL-space X, and by C M the cone generated by M, then all Gateaux differentiability points of X are just Un-s(C M ), and all Frechet differentiability points of X are ∪ int(C M ) (where n-s(C M ) denotes the non-support points set of CM).

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 \ne \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) \ne \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.