Discontinuous homomorphism between Lie groups

Question

I recently began to study Lie group theory and I found the classical result:

Theorem: Every continuous homomorphism $\varphi: \mathcal{G} \to \mathcal{H}$ of groups is a homomorphism of Lie groups.

I was wondering the following: is the continuity of the homomorphism necessary? I was trying to find a discontinuous homomorphism between Lie groups but I'm unable to do so. There a fairly easy/non exotic example?

Answer 1

Take $G,H$ to be the group of positive reals under multiplication. Taking the logarithm transforms this question into finding a function $f: \mathbb{R}\rightarrow \mathbb{R}$ such that $f(a+b)=f(a)+f(b)$ for all real $a,b$ but $f$ is not continuous. This can be done with the Axiom of Choice. See here: Additive function $T: \mathbb{R} \rightarrow \mathbb{R}$ that is not linear..

As for if Choice is necessary to find a counterexample, I would be interested to know as well.