If $f,g$ are entire functions and$\ fg\equiv 0$ then either $f \equiv 0$ or $g\equiv0. $ Let $f,g$ be entire functions such that $g \not\equiv 0.$ If $fg\equiv0$ in $\mathbb{C},$  could anyone advise me how to show $f \equiv0$ in $\mathbb{C} \ ?$  Thank you. 

Suppose there exists $z$ such that $f(z) \ne 0$. Then $f$ is non-zero in some neighbourhood of $z$, so $g$ must be zero in the same neighbourhood. And if an entire function is identically zero in the neighbourhood of any point, it is zero in the whole of $\mathbb C$.
