Let $f$ be an analytic function defined on $\mathbb{C}^2$. Suppose it vanishes on a set of the form $U \times S$, where $U$ is a disk and $S$ is a countable set with an accumulation point. Is it true that $f$ is identically zero ?
I am not so familiar with several complex variables, so I have a doubt. But I'm pretty sure it's true. Am I sane ?
This can be shown by reducing to a single variable case twice. First fix the first argument in $U$ to conclude that $f$ must vanish on $U\times\mathbb{C}$. Then fix the second argument to show that $f$ must vanish identically.