已知函数 $f(x, y)=\left\{\begin{array}{ll}\frac{x^2 y}{x^4+y^2}, & x^2+y^2 \neq 0 \\ 0, & x^2+y^2=0\end{array}\right.$;
证明: (1) $\frac{\partial f}{\partial x}(0,0), \frac{\partial f}{\partial y}(0,0)$ 存在;
(2) $\frac{\partial f}{\partial x}(x, y), \frac{\partial f}{\partial y}(x, y)$ 在点 $(0,0)$ 处不连续;
(3) $f(x, y)$ 在 $(0,0)$ 处不可微.