Question

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

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

Answer 1

Answer: (A), (B)

$f^{\prime }\left(x\right)$ is continuous $\forall x$

We have, $f(x+y)=f\left(x\right)+f\left(y\right)$ and $f\left(x\right)$ is differentiable at
$x=0$

Clearly $f\left(x\right)=kx$ serves our purpose and hence $f\left(x\right)$ is continuous for all $x\in$ and $f^{\prime }\left(x\right)=k=$ constant.