Let ƒ : R → R be a function such that ƒ = for all x and y…

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

Published by: CGP EDU Academic Team

Published on: August 14, 2026

Let ƒ : R → R be a function such that ƒ = for all x and y, and ƒ(0) = 3 and ƒ ′ (0) = 3. Then -

ƒ(x)/x is continuous on R

ƒ(x) is continuous on R

ƒ(x) is bounded on R

None of these

Text Solution

Verified by Experts

The correct answer is:

ƒ(x + h) = ƒ = [ƒ(2x) + ƒ(2h)]

= ƒ (2x) + ƒ(2h) = ƒ(2x) + ƒ(0)

(ƒ is differentiable at 0 so continuous also )

Putting y = 0 in the given equation, we have

ƒ(x) = ƒ =

Hence ƒ(x + h) = ƒ(x)

Since ƒ(x)/x is not defined at x = 0, ƒ(x)/x is not continuous on R. Clearly ƒ(x) need not be bounded on R. e.g. ƒ(x) = x .

ƒ satisfies the given equation but is not bounded on R.

Previous Question Next Question

Get practice questions, solutions, and test series in one place.

Share your experience with this question and solution.

Send your comment, doubt, correction, or feedback to admin.

What students say about this solution

No reviews yet. Be the first to write a review.