If $M_1$ and $M_2$ are compact connected manifolds of dimension $n$, and $f$ is a continuous map from $M_1$ to $M_2$, f is homotopic to a smooth map from $M_1$ to $M_2$.
Seems to be fairly basic, but I can't find a proof. It might be necessary to assume that the manifolds are Riemannian.
It should be possible to locally solve the problem in Euclidean space by possibly using polynomial approximations and then patching them up, where compactness would tell us that approximating the function in finitely many open sets is enough. I don't see how to use the compactness of the target space though.
Best Answer
It is proved as Proposition 17.8 on page 213 in Bott, Tu, Differential Forms in Algebraic Topology. For the necessary Whitney embedding theorem, they refer to deRham, Differential Manifolds.
This Whitney Approximation on Manifolds is proved as Proposition 10.21 on page 257 in Lee, Introduction to Smooth Manifolds.
There you can find even the proof of Whitney embedding Theorem.