已知函数 $f(x, y)=\left\{\begin{array}{l}\left(x^2+y^2\right) \sin \frac{1}{x y}, x y \neq 0 \\ 0, \quad x y=0\end{array}\right.$, 则在点 $(0,0)$ 处
A. $\frac{\partial f(x, y)}{\partial x}$ 连续, $f(x, y)$ 可微
B. $\frac{\partial f(x, y)}{\partial x}$ 连续, $f(x, y)$ 不可微
C. $\frac{\partial f(x, y)}{\partial x}$ 不连续, $f(x, y)$ 可微
D. $\frac{\partial f(x, y)}{\partial x}$ 不连续, $f(x, y)$ 不可微