Let the function $$f: R \rightarrow R$$ be defined by $$f(x) = x^{3} − x^{2} + (x − 1) \sin x$$ and let $$g: R \rightarrow R$$ be an arbitrary function. Let $$fg: R \rightarrow R$$ be the product function defined by $$(fg)(x) = f(x)g(x)$$. Then which of the following statements is/are TRUE?

A

If g is continuous at $$x = 1$$, then fg is differentiable at $$x = 1$$

B

If fg is differentiable at $$x = 1$$, then 𝑔 is continuous at $$x = 1$$

C

If g is differentiable at $$x = 1$$, then 𝑓𝑔 is differentiable at $$x = 1$$

D

If fg is differentiable at $$x = 1$$, then 𝑔 is differentiable at $$x = 1$$