Let $f:X\to S$ be open proper holomorphic map, $X$ a complex manifold and $S$ a Riemann surface. Is it then true that the critical values $C\subset S$ of $f$ are a discrete supset?
So far I only noted that this would be true, if the set of critical points $K\subset X$ was discrete. But I don't know how to prove it. If $X$ was also a Riemann surface, I could apply the identity principle to $f'$, but in the general case I don't know how to proceed.
Edit: I know about Sard's theorem. I don't see why it is strong enough though.