Let
$$f(x,y) = \begin{cases} \frac{x^3 y^3}{x^2 + y^2}, & (x,y) \neq (0,0) \\ 0, & (x,y) = (0,0) \end{cases}$$
Choose the correct statement(s) from below:\\
(A) $f$ is continuous on $\mathbb{R}^2$;\\
(B) $f$ is continuous at every point of $\mathbb{R}^2 \backslash \{(0,0)\}$;\\
(C) $f$ is differentiable at every point of $\mathbb{R}^2 \backslash \{(0,0)\}$;\\
(D) $f$ is not differentiable at $(0,0)$.