Question

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 $f(x)=x g(x)$
for all $x, y \in R$. If $\displaystyle\lim _{x \rightarrow 0} g(x)=1$, 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 $f^{\prime}(1)$ is equal to $1$
• D. The derivative $f^{\prime}(0)$ is equal to $1$

Answer 1

Answer: (A), (B), (D)

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

Answer 2

(A) $f$ is differentiable at every $x \in R$
(B) If $g(0)=1$, then $g$ is differentiable at every $x \in R$
(D) The derivative $f^{\prime}(0)$ is equal to $1$