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

Yes. Suppose $f$ is not identically zero. Thenis a nonempty open set, and $fg=0$ forces $g=0$ on $U$. Harmonic functions satisfy unique continuation: locally they are real parts of holomorphic functions and hence are real analytic, so a harmonic function vanishing on a nonempty open subset of a connected domain vanishes everywhere. Thus $g\equiv 0$. Interchanging $f$ and $g$ proves that one factor must be identically zero.

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