Let's look like the function

Question

Let's look like the function $ f:\mathbb{C}\rightarrow \mathbb{C}$ $f(x+iy)=\sqrt{|xy|}$ fulfills the Cauchy Riemann conditions in (0,0) but is not derivable in origin

Answer 1

For this functions, $u(x,y)=\sqrt{|xy|}$ and $v(x,y)=0$. So, $v_x(0,0)=v_y(0,0)=0$. On the other hand$$(\forall x,y\in\mathbb{R}):u(x,0)=u(0,y)=0$$and therefore $u_x(0,0)=u_y(0,0)=0$. So, yes, $u_x(0,0)=v_y(0,0)$ and $u_y(0,0)=-v_x(0,0)$. Now, prove that $f'(0)$ doesn't exist.

Answer 2

Note that in https://en.wikipedia.org/wiki/Cauchy%E2%80%93Riemann_equations, it is remarked that for $f$ to be holomorphic the CR equations are not sufficient, $f$ should also be real differentiable. But that is not the case here: the function $t \mapsto f(t,t)$ is not differentiable at $0$.