Past exam of the algebra and analysis I for mathematics and philosophy course of the University of Oxford / 2013 / Analysis / 4 / b / Solution

At any $z\ne 0$, take sufficiently small $h$ so $z+h\ne 0$. The complex difference quotient isHence the complex derivative exists at every point of the punctured plane, andThe excluded point is essential to the domain statement: the reciprocal has a simple pole there.