On Gâteaux Differentiability of Pointwise Lipschitz Mappings

Abstract. We prove that for every function f : X → Y , where X is a separable Banach space and Y is a Banach space with RNP, there exists a set A ∈ Ã such that f is Gâteaux differentiable at all x ∈ S(f) \ A, where S(f) is the set of points where f is pointwise-Lipschitz. This improves a result of Bongiorno. As a corollary, we obtain that every K-monotone function on a separable Banach space is Hadamard differentiable outside of a set belonging to C̃; this improves a result due to Borwein and Wang. Another corollary is that if X is Asplund, f : X → R cone monotone, g : X → R continuous convex, then there exists a point in X, where f is Hadamard differentiable and g is Fréchet differentiable.