Let $u(x, y)$ and $v(x, y)$ be differentiable real functions with continuous partial derivatives in a neighbourhood of $z_0$. Prove that $f = u+ iv$ is complex differentiable at $z_0$ if and only if $$\displaystyle\lim_{r\to 0}\frac{1}{\pi r^2}\oint_{C(z_0,r)}f(z)dz=0$$
Where $C(z_0, r)$ is the circle of radius r centered at $z_0$. I think this maybe related to Cauchy formula but the condition seems to point toward Cauchy Riemann equations.