Consider the function $f : \mathbb{R} \rightarrow \mathbb{R}$ defined as
$$f(x) = \begin{cases} \frac{x}{e^x - 1} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$$
Then which one of the following statements is correct?\\
(A) $f$ is not continuous at $x = 0$.\\
(B) $f$ is continuous but not differentiable at $x = 0$.\\
(C) $f$ is differentiable at $x = 0$ and $f'(0) = -\frac{1}{2}$.\\
(D) $f$ is differentiable at $x = 0$ and $f'(0) = \frac{1}{2}$.