Consider the function $f : \mathbb{R}^{2} \longrightarrow \mathbb{R}$ given by
$$f(x, y) = \left(1 - \cos \frac{x^{2}}{y}\right) \sqrt{x^{2} + y^{2}}$$
for $y \neq 0$ and $f(x, 0) = 0$. (The square root is chosen to be non-negative). Pick the correct statement(s) from below:\\
(A) $f$ is continuous at $(0,0)$.\\
(B) $f$ is an open map.\\
(C) $f$ is differentiable at $(0,0)$.\\
(D) $f$ is a bounded function.