Periodic Orbits of Continuous Mappings of the Circle

Let f be a continuous map of the circle into itself and let $P(f)$ denote the set of positive integers n such that f has a periodic point of period n. It is shown that if $1 \in P(f)$ and $n \in P(f)$ for some odd positive integer n then for every integer $m > n$, $m \in P(f)$. Furthermore, if $P(f)$ is finite then there are integers m and n (with $m \geqslant 1$ and $n \geqslant 0$) such that $P(f) = \{ m, 2 m, 4 m, 8 m, \ldots , {2^n} m\}$.