Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 24G / ii

Yes. The setsare open. The equation $fg=0$ makes them disjoint, while the assumption that $f$ and $g$ never vanish simultaneously gives $U\cup V=\mathbb{C}$. Since $\mathbb{C}$ is connected, one of $U,V$ must be empty. Therefore one of $f,g$ is identically zero.