On Fri, 12 Jul 2013 21:05:15 -0700, William Elliot <marsh@panix.com> wrote:

>Let (S,d) be a compact connected metric space and >f:S -> S a homeomorphism. How would one show >there's some distinct x,y with d(f(x),f(y)) = d(x,y)?

You still haven't given a correct statement of the problem! One of the counterexamples quasi mentioned is a counterexample to this version as well.