I am trying to prove the following Lemma:
Suppose $u$: $\mathcal R^N\to \mathbb R$ is $C^1$. Then for each $\epsilon>0$, there exists a $C^{1,\alpha}$ function $\tilde f$ such that $$ \mathcal L^N\{x,\,\,\tilde f(x)\neq f(x)\}\leq \epsilon, $$ for any $0<\alpha<1$.
I am trying to use Whitney's extension theorem but so far I have no luck... Any help is really welcome!