Hi need some help with this problem:
Assume f : \mathbb{R} → \mathbb{R} is a continuous function, satisfying f(α) = f(β) +f(α −β) for any α, β ∈ \mathbb{R}, and f(0) = 0. Then f(α) = α f(1).
any hints, thank you.
Answer
Note that f(-\beta)=f(0-\beta)=f(0)-f(\beta)=-f(\beta) then f(\alpha-\beta)=f(\alpha)+f(-\beta) and also f(\alpha+\beta)=f(\alpha-(-\beta))=f(\alpha)+f(-(-\beta))=f(\alpha)+f(\beta) and this is a "Cauchy functional equation" conditon of continuity.
See the hint of @Nate. or see the book "Kaczor and Nowak - Problems in Mathematical Analysis II (2000)" 1.6.2 exercise and then conclude your proof.
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