Question

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

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

Answer 1

The Correct Option is C

f (x + y) = f (x) + f (y), as f (x) is differentiable at x = 0.
$\Rightarrow f \, ' (0) = k $ $\hspace15mm$ ...(i)
Now, $ f \, ; (x) = lim_{ h \to 0 } \frac{ f (x + h) - f (x)}{ h} $
$\hspace15mm$ = $ lim_{ h \to 0 } \frac{ f (x) + f (h) - f (x)}{ h}$
$\hspace15mm$ $lim_{ h \to 0 } \frac{ f (h)}{ h}$ $\hspace15mm$ $\bigg [ \frac{0}{0} \, form \bigg]$
Given, f (x + y) = f (x) + f (y), $ \forall $ x, y
$\therefore$ f (0) = f (0) + f (0),
when x = y = 0 $\Rightarrow $ f (0) = 0
Using L'Hospital's rule,
$\hspace15mm$ = $ lim_{ h \to 0 } \frac{ f \, ; (h)}{ 1} $ f ' (0) = k
$\Rightarrow$ f ' (x) = k, integrating both sides,
f(x) = k x + C, as f (0) = 0
$\Rightarrow C = 0 $
$\therefore$ f (x) = k x
$\therefore$ f (x) is continuous for all x s R and f ' (x) = k, i.e.
constant for all x $\in$ R
Hence, (b) and (c) are correct.