This is an exercise from Remmert.
Let D be domain in $\mathbb{C} \ $, and $f : D \rightarrow \mathbb{C} \ $ a real-differentiable function. Suppose that for some $ c \in D $ the limit
$\lim_{h \to 0} |\frac{f(c + h) - f(c)}{h}|$
exists. Prove that either $f \ $ or $\bar{f} \ $ is complex-differentiable at c.
Any hint ?