prove the the conjugate of a holomorphic function f is differentiable iff f'=0

This probably has a simple solution, but I cannot find it so far...

Quote:

Originally Posted by **problem statement**

Suppose the complex-valued function $\displaystyle f$ is holomorphic about zero. Show that $\displaystyle g:=\overline{f}$, i.e. the conjugate of $\displaystyle f$, is differentiable at $\displaystyle z$ if and only if $\displaystyle f'(z)=0$.

The first direction follows fairly directly from the Cauchy-Riemann equations. However, I still need to show that $\displaystyle f'(z)=0$ implies $\displaystyle g'(z)$ exists, and I can't seem to make it happen. Any help would be much appreciated.

Thanks!