Given the function $f:\mathbb{R^2}\rightarrow \mathbb{R}$,
$f(x,y) = \frac{xy(x²-y²)}{x²+y²}, $ if $ (x,y) \neq (0,0)$
$f(x,y) = 0, $ if $ (x,y) = (0,0)$
Why isn't valid $\frac{\partial^2}{\partial x \partial y} f(0,0) = \frac{\partial^2}{\partial y \partial x} f(0,0) $ ? Isn't $f$ 2 times differentiable, since I can calculate all the four 2-order partial derivatives of $f$ and since all the limits of them are $0$ and $f(0,0)=0$? Probably I'm not properly evaluating the limits. I believe $f$ can't be 2-times differentiable, since Schawrz Theorem is "failing" for $f$.
Can someone help me? Thanks.