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### IIT-JEE 2011 Paper 1 Offline

MCQ (More than One Correct Answer)

Let f : R $$\to$$ 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.

## Explanation

Set x = 0 in the functional equation to obtain

$$f(0) = f(0) + f(0)$$ $$\therefore$$ $$f(0) = 0$$

$$f'(x) = \mathop {\lim }\limits_{h \to 0} {{f(x + h) - f(x)} \over h}$$

$$= \mathop {\lim }\limits_{h \to 0} {{f(x + h) - f(x + 0)} \over h} = \mathop {\lim }\limits_{h \to 0} {{f(x) + f(h) - f(x) - f(0)} \over h}$$

$$= \mathop {\lim }\limits_{h \to 0} {{f(h) - f(0)} \over h} = f'(0)$$

Thus, $$f'(x) = \lambda$$ (say). Also $$f(x) = \lambda x + \mu$$

As $$f(x) = 0$$ we have $$\mu = 0$$ $$\therefore$$ $$f(x) = \lambda x$$.

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