Show that $f(z)=\sqrt{|xy|}$ is not analytic at $(0,0)$ even though the Cauchy Riemann conditions are satisfied there. If $f(z)=\sqrt{|xy|}$, how do I find the partial derivatives of $u$ and $v$? How can we show it is not analytic at $(0,0)$?

In the real case $\lim_{h\to 0} \frac {|f(h,0)|}{|h|}=0=\frac {|f(0,h)|}{|h|}$ hence if it is differentiable at $0$ the derivative would be $0$ (Since if $f$ where differentiable the derivative would be the partial derivatives in a matrix)  but $\lim_{(h,h)\to 0}\frac{|f(h,h)-f(0,0)-0|}{|(h,h)|}=\frac 1 {\sqrt{2}}$ which mean the limit $\lim_{(h,k)\to 0}\frac{|f(h,k)-f(0,0)-0|}{|(h,k)|}$ is not $0$ which shows $f$ is not real differentiable at $0$. 
