I'm reminded of an exercise in real analysis :

If $f:\mathbb{R} \to\mathbb{R}_{>0}$ is continuous, bounded on $[0,1]$, and satisfies $f(x+y)=f(x)f(y)$, then $f(x) = e^{cx}$ for some constant $c$.

This makes me wonder if something similar should be true in the case of $p$-adic units.

Question : are the continuous characters of the form

• $\eta : \mathbb{Z}_p^* \to \mathbb{Z}_p^*$, or
• $\eta : (1+p\mathbb{Z}_p)^{\times} \to \mathbb{Z}_p^*$ (i.e., on the principal units in $\mathbb{Z}_p^*$)

well understood? For example, does there exist a classification in terms of the $p$-adic logarithm in the second case, or classifications in either case ?