(a) A continuous map $F$ of an interval into itself has a periodic orbit of period 3 . Prove that $F$ also has periodic orbits of period $n$ for all positive integers $n$.

(b) What is the minimum number of distinct orbits of $F$ of periods 2,4 and 5 ? Explain your reasoning with a directed graph. [Formal proof is not required.]

(c) Consider the piecewise linear map $F:[0,1] \rightarrow[0,1]$ defined by linear segments between $F(0)=\frac{1}{2}, F\left(\frac{1}{2}\right)=1$ and $F(1)=0$. Calculate the orbits of periods 2,4 and 5 that are obtained from the directed graph in part (b).

[In part (a) you may assume without proof:

(i) If $U$ and $V$ are non-empty closed bounded intervals such that $V \subseteq F(U)$ then there is a closed bounded interval $K \subseteq U$ such that $F(K)=V$.

(ii) The Intermediate Value Theorem.]