Let $M$ be a complex manifold, and let $f : M \to \mathbb C$ be a continuous function which is holomorphic everywhere but on a closed hypersurface $H \subset M$. Can I deduce that $f$ is holomorphic on all of $M$?
Of course, if $M$ is a Riemann surface, then $H$ is a bunch of isolated points, so the Riemann extension theorem applies. But what about the higher-dimensional case?
The answer is yes. The Weyl lemma says that every distributional solution to the CR equations is actually a holomorphic function. Your $f$ definitely qualifies.