Past exam of the metric spaces and complex analysis course of the University of Oxford / 2014 / 3 / a / Solution

At $z=x+iy$, compute the complex derivative first along real increments and then along imaginary increments. Holomorphy givesEquating real and imaginary parts proves the Cauchy-Riemann equations:The partial derivatives exist because these directional difference quotients are limits of the holomorphic derivative. This argument proves the necessary equations; it does not claim that their pointwise existence alone is sufficient for holomorphy.