On Φ-differentiability of Functions over Metric Spaces

In 1933 S. Mazur [4] proved the following Theorem 1. Let (X, ·) be a separable real Banach space. Let f be a real-valued convex continuous function defined on an open convex subset Ω ⊂ X. Then there is a subset A ⊂ Ω of the first category such that f is Gateaux differentiable on Ω \ A. The result of Mazur was a starting point for the theory of differentiability of convex functions (cf. the book of Phelps [7]). In 1968 Asplund showed that under the assumption that in the conjugate space X * there is an equivalent strictly convex norm (in particular, if X * is separable) we can obtain the Fréchet differentiability. Theorem 2 ([1]). Let (X, ·) be a real Banach space. Suppose that in the conjugate space X * there is an equivalent strictly convex norm. Let f be a real-valued convex continuous function defined on an open convex subset Ω ⊂ X. Then there is a subset A ⊂ Ω of the first category such that f is Fréchet differentiable on Ω \ A. A question arises how to extend this theorem to functions defined on a metric space without any linear structure.