IIT-JEE 2011 · Paper 1 · Mathematics Q57

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MathematicsMultiple correct+4Question 57 of 69

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

f(x)

is differentiable only in a finite interval containing zero

f(x)

is continuous

\forall x\in\mathbb R

f'(x)

is constant

\forall x\in\mathbb R

f(x)

is differentiable except at finitely many points

Answer and worked solution

Official answer: B, C

Cauchy's equation. $f(0)=0$ and
$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}h=\lim_{h\to0}\frac{f(h)}h=f'(0),$
so $f$ is differentiable everywhere with constant derivative: $f(x)=kx$ .

(B) continuous everywhere ✓; (C) $f'$ constant ✓.
(A) is false. (D) is vacuously true (no exceptional points at all), which is why the key also accepts BCD.

(B), (C) (key: BC or BCD)

Topics:Differentiability Continuity

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IIT-JEE 2011 · Paper 1, Mathematics question 57