Hi need some help with this problem:
Assume f:R→R is a continuous function, satisfying f(α)=f(β)+f(α−β) for any α,β∈R, and f(0)=0. Then f(α)=αf(1).
any hints, thank you.
Answer
Note that f(−β)=f(0−β)=f(0)−f(β)=−f(β) then f(α−β)=f(α)+f(−β) and also f(α+β)=f(α−(−β))=f(α)+f(−(−β))=f(α)+f(β) 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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