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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Differentiability via one-sided directional derivatives


Author: Marián Fabián
Journal: Proc. Amer. Math. Soc. 82 (1981), 495-500
MSC: Primary 58C20; Secondary 26B05
MathSciNet review: 612748
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Abstract: Let $ F$ be a continuous mapping from an open subset $ D$ of a separable Banach space $ X$ into a Banach space $ Y$. We show that if the one sided directional derivative $ D_x^ + F(a)$ of $ F$ at $ a$ in the direction $ x$ exists for each $ (a,x)$ from a dense $ {G_\delta }$ subset $ S$ of an open set $ D \times U \subset X \times X$, then $ F$ is Gâteaux differentiable on a dense $ {G_\delta }$ subset of $ D$. Similar results are obtained for Fréchet differentiability when $ X$ is finite-dimensional and for $ {w^ * }$-Gâteaux differentiability.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0612748-2
PII: S 0002-9939(1981)0612748-2
Article copyright: © Copyright 1981 American Mathematical Society