If f(x + y) = f(x) f(y) ∀ x, y ∈ R and f(x) = 1 + g(x) G(x)…

Maths Functions, Limits, Continuity and Differentiability General Subjective Type

Published on: August 14, 2026

If f(x + y) = f(x) f(y) ∀ x, y ∈ R and f(x) = 1 + g(x)

G(x), where g(x) = 0 and G(x) exists. Prove that f(x) is continuous for all x ∈ R.

Text Solution

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The correct answer is:

CHECK THE SOLUTION.

Sol. Let a be any real number, then

f(x) = f(a –h) = f(a) f(–h) = f(a) f(–h)

= f(a) (1 + g (–h) G(–h))

= f(a) { 1 + g(–h) G(–h)}

= f(a) {1 + 0 × a finite number} = f(a)

Similarly = f(a)

∴ f(x) = f(x) = f(a)

⇒ f(x) is continuous ∀ x ∈ R.

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