I'm doing this problem on Stein.
Prove that $f = \sqrt{|x||y|}$ for $x, y \in \mathbb{R}$ is not holomorphic at origin but satisfies the Cauchy-Riemann Equation at $0$.
Write $f = u + iv$. I calculated the derivative at $0$ for $u(x, y) = \sqrt{|x||y|}$ and got:
$\frac{\partial u}{\partial x} = \pm \frac{1}{2}\sqrt{\frac{|y|}{|x|}}$ (deepening on the signs of $x$), and $\frac{\partial v}{\partial y} = 0$ since $f$ is real (thus $v$) is real. So I cannot show that $f$ satisfies the CR at $0$. Am I missing something?
Thanks for any help!
\begin{align*} \dfrac{\partial u}{\partial x}(0,0)&=\lim_{h\rightarrow 0}\dfrac{u(h,0)-u(0,0)}{h}\\ &=\lim_{h\rightarrow 0}\dfrac{0-0}{h}\\ &=0, \end{align*} similar for $\dfrac{\partial u}{\partial y}(0,0)=0$, so the Cauchy-Riemann condition holds.