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Given $f(\frac{x+a}{b})=f(\frac{f(x)+a}{b})$, $x$ is a real number, $a$ is an integer number and $b$ is a natural number.

What are all the functions that satisfying this restriction?

I tried to put some numbers for $a,b$ but can't see how it helps me..

Thanks!

debi
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  • What is the domain and range of the function? R to R? – Sandeep Silwal Apr 22 '14 at 15:50
  • I guess this is true for every $x$ such that $f(x)=x$, i.e. it is true for every fixed points of $f$. – 7raiden7 Apr 22 '14 at 15:58
  • @7raiden7 - but is it just $f(x)=x$ and you can one show it? and thanks for your comment – debi Apr 22 '14 at 18:15
  • If $f$ is constant this is always true. More strictly, if $f$ is injective then its argument on LHS and RHS must be equal, and that leads to my conclusion. So, every injective function with a fixed point satisfies your condition. If $f$ is neither injective nor constant, I guess this would be the trickiest case. – 7raiden7 Apr 23 '14 at 09:02

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Beside $f(x)=x$, you also have $f(x)=x+c$, when $c$ is a real number. But i don't know how to prove it and probably there are more functions.

Arin
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