Hadamard differentiability via G\^ ateaux differentiability

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Abstract

Let $X$ be a separable Banach space, $Y$ a Banach space and $f: X \to Y$ a mapping. We prove that there exists a $\sigma$-directionally porous set $A\subset X$ such that if $x\in X \setminus A$, $f$ is Lipschitz at $x$, and $f$ is G\^ateaux differentiable at $x$, then $f$ is Hadamard differentiable at $x$. If $f$ is Borel measurable (or has the Baire property) and is G\^ ateaux differentiable at all points, then $f$ is Hadamard differentiable at all points except a set which is $\sigma$-directionally porous set (and so is Aronszajn null, Haar null and $\Gamma$-null). Consequently, an everywhere G\^ ateaux differentiable $f: \R^n \to Y$ is Fr\' echet differentiable except a nowhere dense $\sigma$-porous set.