I'm wondering about the following:

Every continuous map between smooth manifolds is homotopic to a smooth map.

By density of polynomials in space of continuous functions on [0,1], continuous functions can be approximated by smooth ones with respect to the maximum norm.

Now, assume $G_1$ and $G_2$ are Lie groups; maybe asume $G_1$ and $G_2$ compact. Let $f:G_1\rightarrow G_2$ be a continuous map. Can $f$ approximated (in some sense) by a continuous group homomorphism?

The answer by Dmitri raises another question.

What classes of continuous functions between Lie groups can be approximated by continuous group homomorphisms?

closedin the set of continuous maps (for the topology of uniform convergence on compact subsets). So continuous homomorphisms only approximate continuous homomorphisms (and, if one allows pointwise convergence, they cannot approximate more than group homomorphisms). $\endgroup$