Let $$\mathbb{R}$$ denote the set of all real numbers. Let $$f:\mathbb{R}\to\mathbb{R}$$ be an arbitrary function and let $$g:\mathbb{R}\to\mathbb{R}$$ be the function defined by

$$g(x)=x\,f(x),\quad\text{for all }x\in\mathbb{R}.$$

Then which of the following statements is (are) TRUE?

A

The function $$g$$ is always continuous at $$x=0$$

B

If $$f$$ is continuous at $$x=0$$, then $$g$$ is differentiable at $$x=0$$

C

If $$g$$ is differentiable at $$x=0$$, then $$f$$ is continuous at $$x=0$$

D

If $$g$$ is differentiable at $$x=0$$, then $$\lim_{x\to 0}f(x)$$ exists