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Proceedings of the American Mathematical Society

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Differentiability via directional derivatives

Authors: Ka Sing Lau and Clifford E. Weil
Journal: Proc. Amer. Math. Soc. 70 (1978), 11-17
MSC: Primary 26A24; Secondary 58C20
MathSciNet review: 0486354
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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.

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Keywords: Baire space, dense $ {G_\delta }$, directional, Gâteaux, Frechet derivatives
Article copyright: © Copyright 1978 American Mathematical Society