Holomorphic functions on a domain: $fg\equiv 0$ implies $f\equiv 0$ or $g\equiv 0$

Show that if $fg \equiv 0$ on a domain $\Omega$ then either $f\equiv 0$ or $g\equiv 0$.

Solution

If $f\not\equiv 0$, there is a point $z_0$ where $f(z_0)\neq 0$, and thus a neighborhood $U\ni z_0$ where $f$ is nonvanishing. This forces $g\equiv 0$ on $U$, however $U$ is a set with a limit point, so $g\equiv 0$ on $\Omega$ by the identity principle.