I was wondering if there is a continuous function such that $f(f(x)) = xf(x)$ for every positive number $x$.
How to find the function $f$ given $f(f(x)) = xf(x)$?
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real-analysis
functional-equations
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0$f(x) = 0$ for $\forall x$ :) – 2012-09-29
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1You must have $f(1)=0$ or $1$. – 2012-09-29
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1Did you mean to require $f(x)$ to be positive as well? – 2012-09-30
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0At the invertible places, the equation also reads $f(x)=x\cdot f^{-1}(x)\ $. – 2012-09-30