Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be functions satisfying $$f(x + y) = f(x) + f(y) + f(x)f(y) \text{ and } f(x) = xg(x)$$ for all $x, y \in \mathbb{R}$. If $\lim_{x \rightarrow 0} g(x) = 1$, then which of the following statements is/are TRUE? (A) $f$ is differentiable at every $x \in \mathbb{R}$ (B) If $g(0) = 1$, then $g$ is differentiable at every $x \in \mathbb{R}$ (C) The derivative $f'(1)$ is equal to 1 (D) The derivative $f'(0)$ is equal to 1
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be functions satisfying
$$f(x + y) = f(x) + f(y) + f(x)f(y) \text{ and } f(x) = xg(x)$$
for all $x, y \in \mathbb{R}$. If $\lim_{x \rightarrow 0} g(x) = 1$, then which of the following statements is/are TRUE?
(A) $f$ is differentiable at every $x \in \mathbb{R}$
(B) If $g(0) = 1$, then $g$ is differentiable at every $x \in \mathbb{R}$
(C) The derivative $f'(1)$ is equal to 1
(D) The derivative $f'(0)$ is equal to 1