
Journal of Convex Analysis 21 (2014), No. 3, 703713 Copyright Heldermann Verlag 2014 Gâteaux and Hadamard Differentiability via Directional Differentiability Ludek Zajícek Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic zajicek@karlin.mff.cuni.cz [Abstractpdf] 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 onesided 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 onesided 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, sigmadirectionally porous set, pointwise Lipschitz mapping. MSC: 46G05; 26B05, 49J50 [ Fulltextpdf (131 KB)] for subscribers only. 