Continuous group homomorphism

Remarks

Example: $f \colon (\R, +) \to (\C \setminus \set{0}, \cdot)$, $f(t) = e^{2\pi i t}$. It satisfies $f(s + t) = f(s) f(t)$, and it is continuous as a composition of continuous functions. Non-example: the identity map from $(\R, +)$ with the usual topology to $(\R, +)$ with the discrete topology. It is a homomorphism, but the preimage of the open set $\set{0}$ is $\set{0}$, which is not open in the usual topology. A homomorphism $f$ is continuous if and only if it is continuous at the identity $e$ of $G$. Indeed, near a point $a$ one writes $f(x) = f(a) f(a^{-1} x)$ and uses that translations are homeomorphisms.