Limits Continuity And Differentiability

Let $f, g: \mathbf{R} \rightarrow \mathbf{R}$ be functions defined by

$$ f(x)=\left\{\begin{array}{cc} x \sin \left(\frac{1}{x}\right), & \text { for } x \neq 0 \\ 0, & \text { for } x=0 \end{array}\right. $$

and $g(x)=x f(x)$

Consider the following statements

(i) $f(x)$ is continuous at $x=0$ but not differentiable at $x=0$

(ii) $g(x)$ is differentiable at $x=0$, but $g^1(x)$ is not continuous at $x=0$

Then, which one of the following is true?