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Maths Functions, Limits, Continuity and Differentiability General Single Correct MCQ

Published by: CGP EDU Academic Team

Published on: August 11, 2026

The function ƒ(x) = [x] 2 – [x 2 ] (where [y] is the greatest integer less than or equal to y), is discontinuous at -

All integers

All integers except 0 and 1

All integers except 0

All integers except 1

Text Solution

Verified by Experts

The correct answer is:

Note that ƒ(x) = 0 for each integral value of x.

Also, if 0 ≤ x < 1, then 0 ≤ x 2 < 1

∴ [x] = 0 and [x 2 ] = 0 ⇒ ƒ(x) = 0 for 0 ≤ x < 1.

Next, if 1 ≤ x < , then

1 ≤ x 2 < 2 ⇒ [x] = 1 and [x 2 ] = 1

Thus, ƒ(x) = [x] 2 – [x 2 ] = 0 if 1 ≤ x < .

It follows that ƒ(x) = 0, if 0 ≤ x < .

This shows that ƒ(x) must be continuous at x = 1.

However, at points x other than integers and not lying between 0 and , ƒ(x) ≠ 0.

Thus, ƒ is discontinuous at all integers except 1.

Hence is the correct answer.

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