It is a theorem of Riemann that if a function $f:D\to\Bbb C$ is holomorphic in all but finitely many points where it is continuous, then in fact $\mathcal O(D)\ni f$. An exercise in Remmert's introductory text asks to prove this is true if $f$ is holomorphic in $D$ except for a line in $D$, where it is continuous. Can one characterize the subsets $C$ of a fixed open set $D$ such that the claim "If $f$ is holomorphic in $D\setminus C$, $f$ is holomorphic in $D$." holds true?