Assuming that $f$ has $C^1$ real and imaginary parts, prove that:
$\displaystyle \lim_{r \to 0} \frac{1}{r^2} \int_{C_r}f(z)dz = 2\pi i\frac{\partial{f}}{\partial{\bar{z}}}(z_0).$
Additionally, show that the above equality implies that $f$ is differentiable at $z_0$ iff the limit on the left hand side vanishes.