Let f : R → R is a real valued function ∀ x, y ∈ R such tha…

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

Published by: CGP EDU Academic Team

Published on: August 13, 2026

Let f : R → R is a real valued function ∀ x, y ∈ R such that | f(x) – f(y)| ≤ |x – y| 2 . The function h(x) = dx is –

Every where continuous

Discontinuous at x = 0 only

Discontinuous at all integral points

h(0) = 0

Text Solution

Verified by Experts

The correct answer is:

Since |f(x) – f(y)| ≤ | x – y | 2 x ≠ y

∴ ≤ | x – y |

Taking lim as y → x, we get

≤ |x – y|

⇒ ≤

⇒ |f ′ (x) ≤ 0| ⇒ |f ′ (x)| = 0 (  |f ′ (x) ≥ 0|)

∴ f ′ (x) = 0 ⇒ f(x) = C (constant)

∴ h(x) = dx = = cx + d where d is constant of integration.

h(x) of a linear function of x which is continuous for all x ∈ R.

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