Hadamard differentiability via Gâteaux differentiability

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 Gâteaux differentiable at $x$, then $f$ is Hadamard differentiable at $x$. If $f$ is Borel measurable (or has the Baire property) and is Gâteaux differentiable at all points, then $f$ is Hadamard differentiable at all points except for a set which is a $\sigma$-directionally porous set (and so is Aronszajn null, Haar null and $\Gamma$-null). Consequently, an everywhere Gâteaux differentiable $f: \mathbb {R}^n \to Y$ is Fréchet differentiable except for a nowhere dense $\sigma$-porous set.