## Find all real monotone fcns f such that f(f(x)) = f(x) + 2x

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### Find all real monotone fcns f such that f(f(x)) = f(x) + 2x

$f(x)$ is a real, monotonous function and for all real $x$: $f(f(x))=f(x)+2x$. The task is to find all the functions with this property. I have succeeded in showing that the function is strictly monotonous, $f(0)=0$ and trying the linear functions $f(x)=ax+b$ found that only $f(x)=2x$ and $f(x)=-x$ are solutions and therefore I guess that there are no other solutions but I can't prove it. Any help would be greatly appreciated.
james_bond

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Joined: Sun May 17, 2009 4:06 pm