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 $d$?