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Is there any function $f$ which would satisfy $f(x)=f(x+1)$ and $f(-1/x)=f(x)$ for every $x$ or at least positive $x$? For the widest possible domains of $x$?

If I could turn this functional equation into differential equations, I could use some approximate analytic method to get the solution.

Thanks in advance.

In a more general case, is the a function $g$ so $ f \left( \frac{ax+b}{cx+d} \right) = g(x)$?

For real $a$, $b$, $c$ and $d$?

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    I think you want to ask: "Find all the function satisfy..." instead of asking "is there any function which satisfy..." since there are many functions satisfying the functional equation you stated, e.g. constant functions.2011-12-16
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    @Jose Garcia: You would want to impose additional conditions on $f$ relevant to your needs. For example, continuity everywhere is too strong, for then only the constant functions work.2011-12-17
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    What you are asking for is a modular form of weight $0$. You may want to look here (http://en.wikipedia.org/wiki/Modular_form) for more details.2012-06-26

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