Let $S\subset R^3$ be a 2d-regular manifold, and $g:S\rightarrow \mathbb R$ a smooth function. Show that for every $p\in S$ there is an open nbd $V\subset \mathbb R^3$ and smooth function $G:V\rightarrow R$ such that $G|_{S\cap V}=g|_{S\cap V}$.
An optional approach: if $X:U\rightarrow S$, $Y:U'\rightarrow S$ are parametrizations, then $X^{-1}$ is locally a restriction of a smooth function from $R^3$. I guess that could help, but I'm not sure how to finish it up.