Differentiability via Directional Derivatives
نویسندگان
چکیده
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 G8 subset A of D and a Gs subset H of X whose closure has nonempty interior, such that for each a E A and each x E H the directional derivative DxF(a) of F at a in the direction x exists, then F is Giteaux differentiable on a dense G8 subset of D. If X is replaced by R , then we need only assume that the n first order partial derivatives exist at each a E A to conclude that F is Frechet differentiable on a dense, G8 subset of D.
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