Past exam of the mathematics course of the University of Cambridge / 2001 / ib / Paper 2 / 7E (Complex Methods)

At a point $z_0=x+iy$, complex differentiability means thatexists and is independent of the direction of approach. Write $f=u+iv$. Restricting to real $h$ gives $f^{\prime }=u_x+i{v}_x$. Restricting to $h=it$, with real $t\to 0$, givesThese directional limits establish existence of the displayed partial derivatives. Equating their real and imaginary parts gives the Cauchy-Riemann equationsThe argument applies at every point of the given open domain; it does not assume continuity of the partial derivatives beforehand.