I learned from several places that in defining a character of a topological group $G$, we often require it to be continuous, i.e. $\omega:G\to \mathbb{C}^{\times}$ is a continuous group homomorphism. This is particularly the case when $G$ is the $p$-adic field $\mathbb{Q}_p$ or the idele group $\mathbb{A}^{\times}_{\mathbb{Q}}$. Is there any significance to the continuity here? Besides, is here $\mathbb{C}^{\times}$ always equipped with the induced topology from $\mathbb{C}$? Or rather, what is the topology of $\mathbb{C}^{\times}$ here?
I know that in the case of $G=\mathbb{Q}_p$, $\mathbb{C}^{\times}$ is given the discrete topology. Yet I am not very clear about the general case. Will someone be kind enough to say something on this? Thank you very much!
Discontinuous homomorphisms between uncountable groups
usually have no proof of existence without axiom of choice
if they exist, can have extremely unfavorable properties, such as not being measurable.
The case of characters of $\mathbb R$ is paradigmatic, with $\log |\omega|$ a solution of $f(x+y)=f(x)+f(y)$.