I am having trouble with the following statement found in a textbook:
"Let $U$ be a connected open set. Let $f$ be a complex analytic function on $U$ and not constant.
Either $f$ is locally constant and equal to $0$ in a neighbourhood of a zero $z_0$, or $z_0$ is an isolated zero."
I just cannot see how this is the case. For instance, isn't the identity function analytic, not isolated, and not locally constant at $z_0 = 0$?
Edit: While the accepted answer to this question is a good proof of why the above statement is true, the actual trouble I ran into was not properly understanding the terminology. I mistook the meaning "isolated" in this case. If this may serve to help anyone in future, I say this: ensure you properly understand the precise meaning of all terminology before trying to consider a proof.