I'm wondering about the following:


1. Every continuous map between smooth manifolds is homotopic to a smooth map.
2. 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?



Continuous map between Lie groups approximation
Let $G\_1=G\_2=U(1)$. Take $f$ such that its image is in an $1/2$-neighborhood of the identity. On the other hand any group homomorphism is either constant or surjective. So there is no chance of approximating continuous maps by homomorphisms.


