Limits Continuity And Differentiability

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?