Suppose $g(z)$ has an isolated singularity at $z=z_0$ and $|\Re[g(z)]| \ge M>0$ for all $z \in \mathbb C-\{z_0\}$. What is the type of singularity of $g$ at $z_0$?
I have a guess it is removable but I could not argue why it can not be a pole though. To rule out essential, I argue with Casorati-Weierstrass theorem.