Prove or disprove: Suppose $\displaystyle f(z)$ and $\displaystyle g(z)$ are analytic functions on an open and connected region $\displaystyle \omega$ and $\displaystyle f(z)g(z)=0 \in \omega$, then either $\displaystyle f(z)=0$ or $\displaystyle g(z)=0$ in $\displaystyle \omega$.

Can I get some help please?