I'd like to prove the cellular approximation theorem using Sard's theorem. The only hard part is the induction step:
Let $f: \mathbb{D}^k\rightarrow Y$ be a map from the closed $k$-disk to a CW-complex, Y, for which $f(S^{k-1})\subseteq Y^{k-1}$ and let $e^j$ be a $j$ dimensional cell in $Y$ where $j>k$ . Then $f$ is homotopic relative to $S^{k-1}$ to a map whose image omits a point in $e^j$.
Clearly if I'm going to use Sard's theorem I need to homotope $f$ to something which is smooth on the preimage of $e^j$ but I'm not really sure how to proceed. For example, I'm not sure under what conditions a contiuous map homotopes to a smooth one. Furthermore even if I could perform the homotopy I'm not sure how to extend it to the rest of the disk. I could use a bump function but I'm not sure how to guarantee that the resulting function will be well behaved on the region where the bump function is descending.