Thursday, August 15, 2019

calculus - a continuous function, satisfying f(α)=f(β)+f(αβ) for any α,βmathbbR




Hi need some help with this problem:



Assume f:RR is a continuous function, satisfying f(α)=f(β)+f(αβ) for any α,βR, and f(0)=0. Then f(α)=αf(1).




any hints, thank you.


Answer



Note that f(β)=f(0β)=f(0)f(β)=f(β) then f(αβ)=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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