SCHWEITZER 1979 Problem 9

Let us assume that the series of holomorphic functions $\sum_{k=1}^{\infty}f_k(z)$ is absolutely convergent for all $z \in \mathbb{C}$ . Let $H \subseteq \mathbb{C}$ be the set of those points where the above sum funcion is not regular. Prove that $H$ is nowhere dense but not necessarily countable.

L. Kerchy