Is there a non-unitary continuous representation of $S^1$?

Question

I'm wondering whether there exist any continuous representations $S^1\rightarrow GL(1,\mathbb{C})$ with image not contained in $U(1)$. I'm aware that any such representation is unitarisable, but wikipedia (https://en.wikipedia.org/wiki/Circle_group#Representations) makes the stronger claim that any such representation must take values in $U(1)$. But I can't think of an easy proof (or counter-example) of this.

Any help would be appreciated.

Answer 1

Any element of finite order must map to a root of unity. And such elements are dense in $S^1$.