Let $$f:R \rightarrow R$$ and $$g:R \rightarrow R$$ be functions satisfying
$$f(x + y)=f(x)+f(y)+f(x)f(y)$$ and $$f(x) = xg(x)$$
for all $$𝑥, 𝑦 \in R$$. If $$\lim_{x \rightarrow 0} g(x) = 1$$, then which of the following statements is/are TRUE?

A

f is differentiable at every $$x \in R$$

B

If g(0) = 1, then g is differentiable at every $$x \in R$$

C

The derivative $$f′(1)$$ is equal to 1

D

The derivative $$f′(0)$$ is equal to 1