Let a function ƒ : R → R satisfy the equation ƒ(x + y) = ƒ(…

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

Published on: August 13, 2026

Let a function ƒ : R → R satisfy the equation ƒ(x + y) = ƒ(x) + ƒ(y) for all x, y. If the function ƒ(x) is continuous at x = 0, then -

ƒ (x) is discontinuous for all x

ƒ (x) is continuous for all positive real x

ƒ (x) is continuous for all x

None of these

Text Solution

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The correct answer is:

Since ƒ(x) is continuous at x = 0,

∴ ƒ(x) = ƒ(0).

Take any point x = a, then at x = a

ƒ(x) = ƒ(a + h)

= [ƒ + ƒ(h)]

[  ƒ(x + y) = ƒ(x) + ƒ(y)]

= ƒ + ƒ(h) = ƒ + ƒ(0)

= ƒ(a + 0) = ƒ

∴ ƒ(x) is continuous at x = a. Since x = a is any arbitrary point, therefore ƒ(x) is continuous for all x.

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