...The question is as follows. Suppose f is a continuous function on R, and
f(f(x))=x for all x in R. Show that f(x) has at least one fixed point i.e.
Since f is continuous, by Extreme Value theorem, it attains a minimum m and a maximum M, for some values of x in R. (you cannot talk about the extremums of f unless you are sure that f is bounded or the domain of f is bounded. you are only told that f is continuous..)
By Intermediate Value Theorem, there exists a c in (f(m),f(M)) such that
f(f(c))=c=f(c), so f has at least one fixed point on R.
Does this work?
Thanks in advance.