Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2008 / 4 / iii / Solution

The set $U$ has two open connected components, the left and right half-planes. On each the function is constant in a neighborhood of every point, so its complex derivative is zero everywhere. Thus $f$ is holomorphic on $U$. A holomorphic function can take different constant values on different components of a disconnected domain.