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. 

Since the Complexes are an integral domain, $f(z)g(z)=0$ implies either $f(z)=0$ or $g(z)=0$. But then one of $f,g$ , say $f$ must have an uncountable number of zeros in $\mathbb C$. But an uncountable subset of $\mathbb C$ has a limit point in $\mathbb C$ . Then the set of zeros of $f$ has a limit point in $\mathbb C$, so that, by the identity theorem, we must have $f==0$. 
