This is an old revision of the document!
$$\gdef\wt \widetilde, \gdef\RM\backslash$
Thm Suppose $M$ is a smooth manifold, $F: M \to \R^k$ is a continuous map. $\delta: M \to \R$ is a positive function. Then, we can find a smooth function $\wt F: M \to \R^k$, such that $|F(x) - \wt F(x)| < \delta(x)$ for all $x \in M$. Furthermore, if $F$ is already smooth on a closed set $A$, we can choose $\wt F = F$ on $A$.
Sketch of the proof: We do it following steps