Past exam of the algebra and analysis II for mathematics and philosophy course of the University of Oxford / 2013 / 4 / a / Solution

A function is holomorphic on an open set when it is complex differentiable at every point of that set. Explicitly, for every $z\in U$, the limitexists as a finite complex number. The limit is independent of the direction in which the complex increment approaches zero; existence of derivatives only along the real axis would not suffice.