Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / 7 / a

Complex differentiability requires the same limit for increments along the real and imaginary directions. Real increments give $f^{\prime }=u_x+i{v}_x$. Imaginary increments giveEquating real and imaginary parts yields the Cauchy-Riemann equationsThey hold throughout the open region where $f$ is holomorphic. Conversely, continuous first real partial derivatives satisfying these equations on an open set ensure holomorphy there.