Show that $f(x + iy) = \sqrt{|x||y|}$ satisfies CR equations at (0, 0) Consider the function $$f(x + iy) = \sqrt{|x||y|}$$
Show that this function satisfies the cauchy-reimann equations at $z = (0, 0)$ but is not differentiable there.
I tried to evaluate
$$
\frac{\partial u}{\partial x} = \frac{\sqrt{|y|}}{2\sqrt{|x||y|}} 
$$
I know that I'm playing fast and loose with absolute values here, but the problem is that even restricting this to $$(\mathbb R^+, \mathbb R^+)$$ causes the evaluation at 0 to not exist. 
So, I was wondering how to 
a. Evaluate $\frac{\partial u}{\partial x}$ at $(0, 0)$ and 
b. show that the derivative does not exist

Hints:  To show the CR equations hold at $0,$ the formula is just a distraction. Just notice $f$ vanishes on both the $x$ and $y$ axes. For the second part, we do need the formula. If $f'(0)$ exists, then $f'(0) = 0$ (just by looking at the axes again). So then we would have $\lim_{t\to 0} f(t+it)/(t+it) = 0$ as well. Is that true?
