We have been given $f(x,y) = -xy$ if $|y| \ge |x|$ and $xy$ if $|y| < |x|$. We need to show that both $f_{xy}$ and $f_{yx}$ exist at $(0,0)$, but both are unequal.
When I solve this problem in both the regions, I found both $f_{xy}$ and $f_{yx}$ to be $-1$ in first region and $1$ in second. But I needed to show that they exist and are unequal!
The definition of partial derivative I used was:
$$\left.\frac{\partial f(x,y)} { \partial x}\right|_{(a,b)} = \lim_{u\to 0}\frac{f(a+u, b)-f(a,b)}{u}$$