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?

(In reply to Curious by JLo)

Yeah, I think I forgot to add s back in.

Well, I was thinking about a circle and thinking "what maps the reals to a circle" but arctan maps to a half-circle so it just seemed like I must have missed something.

Posted by Joel on 2006-11-09 16:38:51