Question

Let f:RtoR be a function such that f(x+y)=f(x)+f(y). If f(x) is differentiable at x=0, then which one of the following is incorrect?

• A. \(f(x)\) is continuous, \(\forall x\in\mathbb{R}\)
• B. \(f'(x)\) is constant, \(\forall x\in\mathbb{R}\)
• C. \(f(x)\) is differentiable, \(\forall x\in\mathbb{R}\)
• D. f(x) is differentiable only in a finite interval containing zero

Hint:

Cauchy functional equation + differentiability ⟹ linear function.

Answer 1

The Correct Option is D

Additivity with differentiability at zero implies f(x)=kx. Hence f(x) is differentiable for all real x.