Question

Let $f:R\to R$ be a function such that $f(x+y)=f\left(x\right)+f\left(y\right),\forall x,y\in R.$
If $f\left(x\right)$ is differentiable at $x=0$, then

• A. $f\left(x\right)$ is differentiable only in a finite interval containing zero
• B. $f\left(x\right)$ is continuous $\forall x\in R$
• C. $f^{\prime }\left(x\right)$ is constant $\forall x\in R$
• D. $f\left(x\right)$ is differentiable except at finitely many points

Answer 1

Answer: (B), (C)

The correct options are
B $f\left(x\right)$ is continuous $\forall x\in R$
C $f^{\prime }\left(x\right)$ is constant $\forall x\in R$

Given that $f(x+y)=f\left(x\right)+f\left(y\right)$ and $f\left(x\right)$ is differentiable at $x=0$.

Therefore, $f\left(x\right)$ is continuous $\forall x\in R$

We know that $f{}^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{f(x+h)-f\left(x\right)}{h}$

$⟹f{}^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{f\left(x\right)+f\left(h\right)-f\left(x\right)}{h}$

$⟹f{}^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{f\left(h\right)}{h}$ independent of $x$

Therefore, $f^{\prime }\left(x\right)$ is constant $\forall x\in R$