Differentiability Via Directional Derivatives

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 <!-- MATH ${G_\delta }$ --> subset A of D and a <!-- MATH ${G_\delta }$ --> subset H of X whose closure has nonempty interior, such that for each and each the directional derivative of F at a in the direction x exists, then F is Gâteaux differentiable on a dense <!-- MATH ${G_\delta }$ --> subset of D. If X is replaced by , then we need only assume that the n first order partial derivatives exist at each to conclude that F is Frechet differentiable on a dense, <!-- MATH ${G_\delta }$ --> subset of D.