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?
Continuous map of Lie Groups
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representation-theory
lie-groups
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2@ThomasAndrews: It's even worse in some cases. For compact Lie groups with finite centers, homomorphisms are automatically continuous, hence automatically smooth. – 2012-03-24