Q:

Cauchy- Riemann Proof

If ${f= u+iv}$ is defined around a given point ${z_0=x_0+iy_0}$, and ${ \lim_{h\to0} Re (\frac{f( z_0+h)-f(z_0)}{h}) }$ exists, prove that ${u_x}$ and ${v_y}$ exist and that ${u_x=v_y}$.
From the limit we can understand that ${u}$ is differentiable around ${z_0}$ and therefore ${u_x}$ exists, but it is not given that ${f= u+iv}$ is continuous nor  differentiable so we cannot use Cauchy-Riemann to prove that ${u_x=v_y}$!  I don't know what am I missing but this is supposed to be the full question and I have to prove it, any hints?  

A:

Hint: the limit as $h \to 0$ through real values must be equal to the limit as $h \to 0$ through purely imaginary values.  This immediately gives you $u_x=v_y$. 

