Under what conditions does $f(z)$ continuous imply $\arg(f(z))$ continuous?

Question

My question is as stated in the title. Just to restate:

Under what conditions does $f(z)$ continuous imply $\arg(f(z))$ continuous?

Attempt:

I propose that if for all $z$ in the domain of $f$, $ \operatorname{Re}(f(z)) >0 $, then $f$ continuous implies $\arg(f(z))$ continuous...

Answer 1

If f is continuous and never zero, then a continuous log (hence continuous arg as that can be taken as the imaginary part of the log) exists in any simply connected domain where f is defined.