Is the case where the zeros of $f$ or $g$ are isolated possible?

Is the case where the zeros of $f$ or $g$ are isolated possible?

Let us consider the following equation in $\mathbb{C}$
$$f(s)g(s)=0$$
Assume that $f,g:\mathbb{C}¨\mathbb{R}$, then they are not analytic, but
they are probably continuous in some subdomains of $\mathbb{C}$.
My question is: Is the case where the zeros of $f$ or $g$ are isolated
possible?
If yes, then there is a contradiction since both functions are continuous
and one of these functions has a continuum of zeros.