This is an exercise from Remmert's Theory of Complex functions.
Let $D\subset \mathbb{C}$ be a domain and $f:D\rightarrow \mathbb{C}$ a real-differentiable function. Assume that the following limit exists:
$ \mathrm{lim}_{h\rightarrow 0} \left| \frac{f(c+h) - f(c)}{h} \right|.$
Show that either $f$ or $\overline{f}$ is complex-differentiable.
I've tried showing that $\frac{\partial f}{\partial \overline{z}} = 0$ or $\frac{\partial \overline{f}}{\partial z} = 0$ by using the fact that there exist continuous functions $g$ and $h$ such that in $D$ one can write
$f(z) = f(c) + (z-c)g(z) + (\overline{z} - \overline{c})h(z)$
and that $g(c)= f_{z}(c)$ and $h(c) = f_{\overline{z}}$ and then plugging this into the limit above. Does this approach works and I just can´t see how to do it? Can someone give a hint or a guideline solution to this?