I want to prove that if $f$ is an analytic function for which $\Im f$ is constant, this implies that $f$ itself is constant.
So to start off, it's not given that the function is entire or anything, ruling out Liouville. Rather, I'm guessing the maximum modulus theorem will prove useful here (?).
If I were to show using that the real part of $f$ was constant, proving the statement would be easy (just take the absolute value and investigate $e^f$), but in this case I'm none the wiser from that. Should I use the same parametrization $e^f$ and apply the Cauchy-Riemann equations somewhere? Any suggestions welcome.