Order the natural numbers as follows:
Now let be a continuous function from the reals to the reals and suppose in the above ordering. Then if has a point of least period , then also has a point of least period .
A special case of this general result, also known as Sharkovsky's theorem, states that if a continuous real function has a periodic point with period 3, then there is a periodic point of period for every integer .
A converse to Sharkovsky's theorem says that if in the above ordering, then we can find a continuous function which has a point of least period , but does not have any points of least period (Elaydi 1996). For example, there is a continuous function with no points of least period 3 but having points of all other least periods.
Sharkovsky's theorem includes the period three theorem as a special case (Borwein and Bailey 2003, p. 79).