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

Updated On: Jun 14, 2022
  • $f (x)$ is differentiable only in a finite interval containing zero
  • $f(x)$ is continuous $\forall x \in R$
  • $f ( x )$ is constant $\forall x \in R$
  • $f (x)$ is differentiable except at finitely many points
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The Correct Option is C

Solution and Explanation

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.
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Concepts Used:

Continuity

A function is said to be continuous at a point x = a,  if

limx→a

f(x) Exists, and

limx→a

f(x) = f(a)

It implies that if the left hand limit (L.H.L), right hand limit (R.H.L) and the value of the function at x=a exists and these parameters are equal to each other, then the function f is said to be continuous at x=a.

If the function is undefined or does not exist, then we say that the function is discontinuous.

Conditions for continuity of a function: For any function to be continuous, it must meet the following conditions:

  • The function f(x) specified at x = a, is continuous only if f(a) belongs to real number.
  • The limit of the function as x approaches a, exists.
  • The limit of the function as x approaches a, must be equal to the function value at x = a.