Let $f: R \rightarrow R$ be a function such that $f\left( {\frac{{x + y}}{3}} \right) = \frac{{f(x) + f(y)}}{3}\,\,{\rm{ for}}\,\,{\rm{ all }}\,\,x,y \in R$ $f(0)=0$ and $f^{\prime}(0)=3$. Then, $f(x)$ is

continuous but not differentiable

JEE MATHEMATICS MCQ: DIFFERENTIATION

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Analyzing +4 | -1

Let

f: R \rightarrow R

be a function such that

f\left( {\frac{{x + y}}{3}} \right) = \frac{{f(x) + f(y)}}{3}\,\,\mathrm{ for}\,\,\mathrm{ all }\,\,x,y \in R

f(0)=0

and

f^{\prime}(0)=3

. Then,

f(x)

View answer and explanation

Correct option: C

differentiable in

R

The correct answer is C (differentiable in

R

) because the given functional equation and differentiability at 0 imply that

f(x)

is a linear function, hence differentiable everywhere.