Complex Analysis: Nowhere complex differentiable

PROBLEM

Show that $f(z)=Re(z)$ is nowhere complex differentiable.

PROOF

Let $h=\Delta x+i\Delta y$. Then $$\lim_{y\rightarrow 0}\frac{Re(z+h)-Re(z)}{h}=\lim_{y\rightarrow 0}\frac{Re(h)}{h}$

which is 1 if $h=\Delta x$ and 0 if $h=i\Delta y$.