Let $U \subset \mathbb{R}^n$ be open, let $K \subset\subset U$, and let $f \in C^1(U;\mathbb{R}^m)$. Fix a standard compactly supported mollifier $\eta \in C_c^\infty(B(0,1))$ with $\eta \geq 0$ and
at every $x \in U$ for which $B(x,\varepsilon)\subset U$. Then, for all sufficiently small $\varepsilon>0$, the map $f_\varepsilon$ is smooth on a neighbourhood of $K$, and