Originally Posted by

**arsenicbear** Hi I would just like for someone to check my proof/point out anything I am missing. Thanks!

Prove that the function$\displaystyle f(z)= \sqrt{|xy|}$ is not diff at origin.

Proof:

If it was differentiable at the origin, then $\displaystyle f(z) $ should have the same derivative no matter how you approach the origin. We consider two paths approaching the origin from the x axis (positive side) and the $\displaystyle x=y$ line from the 1st quadrant.

Case 1 (x-axis):

$\displaystyle \lim_{(x,0)\to(0,0)}\frac{f(x,0)-f(0,0)}{x} = \lim_{(x,0)\to(0,0)}\frac{0}{x}$

Case 2 (x=y line):

$\displaystyle \lim_{(x,x)\to(0,0)}\frac{f(x,x)-f(0,0)}{(x,x)} = \lim_{(x,x)\to(0,0)}\frac{x-0}{x+ix}= \lim_{(x,x)\to(0,0)}(\sqrt{2}/2-\sqrt{2}/2*i)$

Since these two limits are different. The function is not differentiable at the origin.