Question

Let f:R → R be a function such that f(x + y) = f(x) + f(y) for all x, y in R. If f(x) is differentiable at x = 0, then

(a) f(x) is differentiable only in a finite interval containing zero.

(b) f(x) is continuous " x ∈ R

(c) is constant " x ∈ R

(d) f(x) is differentiable except at finitely many points.

Answer 1

Correct option (b, c)

Explanation:

Given f(x + y) = f(x) + f(y)

Put x = 0 = y, f(0) = 0

On integration, we get, f(x) = kx + c

If x = 0, f(0) = 0, then c = 0.

Thus, f(x) = kx.

⇒ f'(x) = k