The identity as a third iterate

mark
Mon, 16 Oct 2017 13:44:24 +0000
Problems

Here's a fun problem that you can solve by applying Sharkovsky's theorem.

Suppose that [imath]f:\mathbb R \to \mathbb R[/imath] is a continuous function such that
[dmath]f\circ f \circ f(x) = x[/dmath]
for all [imath]x[/imath]. That is, [imath]f^3[/imath] is the identity function. Prove that [imath]f(x)=x[/imath].