Cartan's theorem on continuous homomorphisms

Remarks

The graph $\Gamma = \set{(g, \varphi(g)) : g \in G}$ is a subgroup of $G \times H$, and it is closed since $\varphi$ is continuous. By the closed subgroup theorem, $\Gamma$ is an embedded Lie subgroup. The projection $\Gamma \to G$ is a bijective smooth homomorphism; such a map has constant rank, so it is a diffeomorphism, and $\varphi$ is its inverse followed by the smooth projection $\Gamma \to H$. Example: a continuous homomorphism $\varphi \colon \R \to \R$ satisfies $\varphi(x + y) = \varphi(x) + \varphi(y)$ and is therefore $\varphi(x) = cx$, which is smooth. Continuity is necessary: using a basis of $\R$ as a vector space over $\Q$, there are additive maps $\R \to \R$ that are not continuous, hence not smooth.

Going deeper

For a student

A direct consequence is that a topological group carries at most one smooth structure making it a Lie group. Indeed, the identity map between two such structures is a continuous homomorphism, so it is a diffeomorphism. This is why one can speak of "the" Lie group structure of a group such as $SO(n)$. It also means continuous representations of Lie groups on finite-dimensional spaces are automatically smooth, so their differentials give representations of the Lie algebra. The question of which topological groups admit a Lie group structure at all is Hilbert's fifth problem.