Let be a function such that for all

MATHEMATICS · Multi correct · CBSE · Mathematics · Relations and Functions · Relations and Functions

Problem

Let

f:\mathbb{R}\to\mathbb{R}

be a function such that

f(x+y)=f(x)+f(y)

for all

x,y\in\mathbb{R}

. If

f(x)

is differentiable at

x=0

, then (1)

f(x)

is differentiable only in a finite interval containing zero (2)

f(x)

is continuous for all

x\in\mathbb{R}

(3)

f'(x)

is constant for all

x\in\mathbb{R}

(4)

f(x)

is differentiable except at finitely many points