--- title: "A Remarkable Property of Real-Valued Functions on Intervals of the Real Line" url: https://daily.dev/posts/a-remarkable-property-of-real-valued-functions-on-intervals-of-the-real-line-uvx6ke8pe source_url: https://theorylunch.wordpress.com/2019/10/17/a-remarkable-property-of-real-valued-functions-on-intervals-of-the-real-line type: article source: "Planet Haskell" published: 2026-05-22T17:00:30.374Z updated: 2026-05-22T17:00:52.793Z tags: ["algorithms", "math"] reading_time: 6 upvotes: 0 comments: 0 language: en --- > ## Documentation Index > Fetch the complete documentation index at: https://daily.dev/llms.txt > Use this file to discover all available pages before exploring further. # A Remarkable Property of Real-Valued Functions on Intervals of the Real Line **[Planet Haskell](https://daily.dev/sources/phaskell)** · 6 min read · 0 upvotes · 0 comments ## Summary Sharkovsky's theorem is presented in two parts. The 'little' theorem states that a continuous function on a real interval with a periodic point of least period 3 must have periodic points of every least period. The proof uses only the intermediate value theorem and two lemmas about compact intervals. The 'great' theorem generalizes this via the Sharkovsky ordering — a total ordering on positive integers — showing that if a continuous function has a point of least period n, it must have points of every least period m where m comes after n in the Sharkovsky ordering. The ordering places odd numbers first, then their doubles, quadruples, etc., with powers of 2 in reverse order at the end. ## Full article daily.dev links to this article rather than hosting it. Read it at the original source: ## Similar posts on daily.dev - [Notes on the Fourier Transform](https://daily.dev/posts/notes-on-the-fourier-transform-go3dssqrp) · Eli Bendersky · 0 upvotes · 0 comments - [The Universe of Discourse : There are two kinds of theorems](https://daily.dev/posts/the-universe-of-discourse-there-are-two-kinds-of-theorems-jzgjke4u0) · Planet Haskell · 0 upvotes · 0 comments - [The Sylvester–Gallai Theorem](https://daily.dev/posts/the-sylvester-gallai-theorem-zdlvbrk0e) · Hacker News · 0 upvotes · 0 comments - [The Universe of Discourse](https://daily.dev/posts/the-universe-of-discourse-xbzwvstha) · Planet Haskell · 0 upvotes · 0 comments --- Tags: [#algorithms](https://daily.dev/tags/algorithms), [#math](https://daily.dev/tags/math) [View this post on daily.dev](https://daily.dev/posts/a-remarkable-property-of-real-valued-functions-on-intervals-of-the-real-line-uvx6ke8pe)