Hadamard differentiability via G\^ ateaux differentiability

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 $\mathrm{\Gamma }$-null). Consequently, an everywhere G^ ateaux differentiable $f:{\mathbb{R}}^n\to Y$ is Fr' echet differentiable except a nowhere dense $\sigma$-porous set.