Let $f: R \rightarrow R$ be a function such that $f(x + y) = f(x) + f(y){\rm{ }}\,{\rm{for}}\,{\rm{ all }}\,x,y \in R$ If $f(x)$ is differentiable at $x=0$, then which one of the following is incorrect?

$f(x)$ is continuous for all $x \in R$

$f^{\prime}(x)$ is constant for all $x \in R$

JEE MATHEMATICS MCQ: DIFFERENTIATION

1 Easy

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Analyzing +4 | -1

Let

f: R \rightarrow R

be a function such that

f(x + y) = f(x) + f(y)\mathrm{ }\,\mathrm{for}\,\mathrm{ all }\,x,y \in R

f(x)

is differentiable at

x=0

, then which one of the following is incorrect?

View answer and explanation

Correct option: D

f(x)

is differentiable only in a finite interval containing zero

Option D (

f(x)

is differentiable only in a finite interval containing zero) is correct because the given condition implies that

f(x)

is differentiable for all

x \in R

, and

f'(x)

is constant for all

x \in R