Center of MassHard

Question

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

Options

A.f (x) is differentiable only in a finite interval containing zero
B.f (x) is continuous ∀ x ∈ R
C.f′ (x) is constant ∀ x ∈ R
D.f (x) is differentiable except at finitely many points

Solution

f (0) = 0 and f′(x) =
⇒ = f′(0) = k (say)
⇒ f (x) = kx + c ⇒ f (x) = kx ( f (0) = 0).

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