IMC 1998 · Problem 3

Let $0 < c < 1$ and

$$f(x) = \begin{cases} \dfrac{x}{c} & \text{for } x \in [0,c], \\[6pt] \dfrac{1-x}{1-c} & \text{for } x \in [c,1]. \end{cases}$$

We say that $p$ is an $n$-periodic point if

$$\underbrace{f(f(\dots f(p)))}_{n} = p$$

and $n$ is the smallest number with this property. Prove that for every $n \ge 1$ the set of $n$-periodic points is non-empty and finite.