Differentiability via directional derivatives

Abstract:

Let F be a continuous function from an open subset D of a separable Banach space X into a Banach space Y. We show that if there is a dense ${G_\delta }$ subset A of D and a ${G_\delta }$ subset H of X whose closure has nonempty interior, such that for each $a \in A$ and each $x \in H$ the directional derivative ${D_x}F(a)$ of F at a in the direction x exists, then F is Gâteaux differentiable on a dense ${G_\delta }$ subset of D. If X is replaced by ${R^n}$, then we need only assume that the n first order partial derivatives exist at each $a \in A$ to conclude that F is Frechet differentiable on a dense, ${G_\delta }$ subset of D.