JEE Advanced 2026 · Paper 1 · Mathematics Q7

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MathematicsMultiple correct+4 / -1Question 7 of 48

Let $\mathbb R$ denote the set of all real numbers. Let $f:\mathbb R\to\mathbb R$ be an arbitrary function and let $g:\mathbb R\to\mathbb R$ be the function defined by
$g(x)=xf(x),\quad\text{for all }x\in\mathbb R.$
Then which of the following statements is (are) TRUE ?

The function

g

is always continuous at

x=0

f

is continuous at

x=0

, then

g

is differentiable at

x=0

g

is differentiable at

x=0

, then

f

is continuous at

x=0

g

is differentiable at

x=0

, then

\displaystyle\lim_{x\to0}f(x)

exists

Answer and worked solution

Official answer: B, D

(A) Take $f(x)=1/x^2$ for $x\ne0$ , $f(0)=0$ . Then $g(x)=1/x$ near 0, not continuous. False.

(B) $\dfrac{g(x)-g(0)}{x}=f(x)\to f(0)$ when $f$ is continuous at 0, so $g'(0)=f(0)$ exists. True.

(D) If $g'(0)$ exists, then $\displaystyle\lim_{x\to0}f(x)=\lim_{x\to0}\frac{g(x)-g(0)}{x}=g'(0)$ exists. True.

(C) The limit exists, but $f(0)$ can be anything: $f(x)=0$ for $x\ne0$ , $f(0)=1$ gives $g\equiv0$ , differentiable, while $f$ is discontinuous. False.

Topics:Differentiability Continuity

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JEE Advanced 2026 · Paper 1, Mathematics question 7