# A Remarkable Property of Real-Valued Functions on Intervals of the Real Line

Today the 17 October 2019 I discussed a very remarkable fixed point theorem discovered by the Ukrainian mathematician Oleksandr Micholayovych Sharkovsky.

We recall that a periodic point of period $n\geq1$ for a function $f:X\to{X}$ is a point $x_n$ such that $f^n(x_n)=x_n$. With this definition, a periodic point of period $n$ is also periodic of period $m$ for every $m$ which is a multiple of $n$. If $f^n(x_n)=x_n$ but $f^k(x_n)\neq{x_n}$ for every $k$ from 1 to $n-1$, we say that $n$ is the least period of $x_n$.

Theorem 1. (Sharkovsky’s “little” theorem) Let $I\subseteq\mathbb{R}$ be an interval and let $f:I\to\mathbb{R}$ be a continuous function su. If $f$ has a point of least period 3, then it has points of arbitrary least period; in particular, it has a fixed point.