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Question:

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

f'(x) is constant ∀ x ∈ R

f(x) is differentiable except at finitely many points

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

f(x) is continuous ∀ x ∈ R

Solution:

Given that f(x+y) = f(x) + f(y) and f(x) is differentiable at x=0.
Therefore, f(x) is continuous ∀ x ∈ R
We know that f'(x) = limh→0 (f(x+h)-f(x))/h
⇒ f'(x) = limh→0 (f(x) + f(h) - f(x))/h
⇒ f'(x) = limh→0 f(h)/h independent of x
Therefore, f'(x) is constant ∀ x ∈ R