It is instructive to see an example where Hartogs' theorem fails in one dimension. Take and let The function is holomorphic in but cannot be extended to

To understand the example and failure of the theorem it is important to understand the proof. In the proof, the way we construct an extension is that we start with a function holomorphic in modify it in a neighbourhood of to be zero, hence extending as a smooth function through Then we solve the operator inhomogeneous equation to “correct” our extension to be holomorphic. The key point is that has compact support allowing us to solve the equation and find a with compact support. This fails in dimension 1. While we always get a solution the solution can never have compact support. Hence, if we tried the proof with the new function we obtain in the proof does not agree with on any open set and hence is not an extension.