关于函数 $f(x, y)=\left\{\begin{array}{l}x y, x y \neq 0 \\ x, y=0 \\ y, x=0\end{array}\right.$ 给出下列结论
(1) $\left.\frac{\partial f}{\partial x}\right|_{(0,0)}=1$
(2) $\left.\frac{\partial^2 f}{\partial x \partial y}\right|_{(0,0)}=1$
(3) $\lim _{(x, y) \rightarrow(0,0)} f(x, y)=0$
(4) $\lim _{y \rightarrow 0} \lim _{x \rightarrow 0} f(x, y)=0$
其中正确的个数为
A. 4
B. 3
C. 2
D. 1