Continuous map of Lie Groups

Question

This might be a dumb question, but are all continuous maps between Lie groups also homomorphisms? I can only seem to think of examples in which they are (i.e., $GL(n,\mathbb{R}) \to \mathbb{R}$ via the determinant, the covering space map from $\mathbb{R} \to S^1$,...). Conversely, a Lie group homomorphism is defined as a homomorphism that is smooth, so what are some examples of homomorphisms between Lie groups that are not smooth?