Let's call a real-valued continuous function an n-th identity root when it generates the identity function after exactly n compositions with itself. For example f(x)=-x is a 2nd identity root because f(f(x))=x.

1. The function f(x)=1-1/x is a 3rd identity root. Unfortunately it is undefined at x=0. Are there identity roots for n>2 which are defined for all real numbers?

2. For a given real number c and n>1, give an example of an n-th identity root which is defined for all real numbers except c. How many such roots exist?

Well done, Richard and Joel! This is pretty much the solution I found when I used my "circle"-approach. Richard's is more or less a simplified version of Joel's (although I am thinking that one has a typo). The pole can be shifted easily, because if f(x) is a solution then f(x-s)+s is a solution too.

Just curious: How did you come up with the solution if you did not use the map-to-circle trick I suggested?

Posted by JLo on 2006-11-09 15:39:48