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Journal of Convex Analysis 21 (2014), No. 3, 703--713
Copyright Heldermann Verlag 2014

G‚teaux and Hadamard Differentiability via Directional Differentiability

Ludek ZajŪcek
Faculty of Mathematics and Physics, Charles University, SokolovskŠ 83, 186 75 Praha 8, Czech Republic


Let $X$ be a separable Banach space, $Y$ a Banach space and $f: X \to Y$ an arbitrary mapping. Then the following implication holds at each point $x\in X$ except a $\sigma$-directionally porous set:\ If the one-sided Hadamard directional derivative $f'_{H+}(x,u)$ exists in all directions $u$ from a set $S_x \subset X$ whose linear span is dense in $X$, then $f$ is Hadamard differentiable at $x$. This theorem improves and generalizes a recent result of A. D. Ioffe, in which the linear span of $S_x$ equals $X$ and $Y = \mathbb{R}$. An analogous theorem, in which $f$ is pointwise Lipschitz, and which deals with the usual one-sided derivatives and G\^ ateaux differentiability is also proved. It generalizes a result of D. Preiss and the author, in which $f$ is supposed to be Lipschitz.

Keywords: Gateaux differentiability, Hadamard differentiability, directional derivatives, Hadamard directional derivatives, sigma-directionally porous set, pointwise Lipschitz mapping.

MSC: 46G05; 26B05, 49J50

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