Holomorphic function has at most countably zeros Let $f$ be holomorphic in a domain $U$, then $f$ has at most countably zeros. How can I prove this statement? I have seen it in Sheng Gong explicitly, but without any indications.

If the set $\\{z\in U|f(z)=0\\}$ were uncountable, then it would have an accumulation point (i.e. not be discrete), and thus you would have $f=0$ by the identity theorem.