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?

• A

  (i) is true; but (ii) is false
• Correct answer
  B

  Both (i) and (ii) are true
• C

  (i) is false, but (ii) is true
• D

  Both (i) and (ii) are false

Hence, the correct answer is option: 'B'.