If $f,g$ are entire functions and$\ fg\equiv 0$ then either $f \equiv 0$ or $g\equiv0. $

Solution 1:

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 $\mathbb C$.

Solution 2:

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 $f==0$.