On equality of the $L^{\mathrm{\infty }}$ norm of the gradient under the Hausdorff and Lebesgue measure

Let $\mathrm{\Omega }$ be an open subset of ${\mathbb{R}}^n$, and let $f:\mathrm{\Omega }\to \mathbb{R}$ be differentiable ${\mathcal{H}}^k$-almost everywhere, for some nonnegative integer $k<n$, where ${\mathcal{H}}^k$ denotes the $k$=dimensional Hausdorff measure. We show that $\mathrm{\parallel }\mathrm{\nabla }f{\mathrm{\parallel }}_{L^{\mathrm{\infty }}({\mathcal{H}}^k)}=\mathrm{\parallel }\mathrm{\nabla }f{\mathrm{\parallel }}_{L^{\mathrm{\infty }}({\mathcal{H}}^n)}\mathrm{.}$ We deduce that convergence in the Sobolev space $W^{1,\mathrm{\infty }}$ preserves everywhere differentiability. As a further corollary, we deduce that the class $C^1(\mathrm{\Omega })$ of continuously differentiable functions is closed in $W^{1,\mathrm{\infty }}(\mathrm{\Omega })$.