Math 2.0

Real functions Theorem

A monotone function has countably many discontinuities

Builds on One-sided limits of a monotone function existJump discontinuity

Statement

Let II be an interval and let f ⁣:I→ℝf \colon I \to \mathbb{R} be monotone. Then the set of points of II where ff is not continuous is countable.

Remarks

Assume ff increasing. By the existence of one-sided limits, each discontinuity cc inside II gives an open interval (f(c−),f(c+))(f(c^-), f(c^+)). These intervals are disjoint. Each contains a rational number, so there are only countably many of them. All these discontinuities are jumps. Mathlib states it as Monotone.countable_not_continuousAt.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionReal sequence, definition, no evidence levelReal sequencedefinitionOne-sided limit, definition, no evidence levelOne-sided limitdefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionJump discontinuity, definition, no evidence levelJump discontinuitydefinitionMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementOne-sided limits of a monotone function exist, statement, no evidence levelOne-sided limits of a…statementA monotone function has countably many discontinuities, statement, no evidence levelA monotone function h…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Limit point of a set definition
  2. Sequence definition
  3. Limit of a function definition
  4. Real sequence definition
  5. One-sided limit definition
  6. Bounded sequence definition
  7. Limit of a sequence definition
  8. Monotone sequence definition
  9. Jump discontinuity definition
  10. Monotone convergence theorem for sequences statement
  11. One-sided limits of a monotone function exist statement
  12. A monotone function has countably many discontinuities statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Monotonic_function, Some basic applications and results
About this page
Tag0WYTM4C9QG
URIm2://0WYTM4C9QG
Aliasesfroda-theorem
Kindstatement, theorem
WritingW0 stub
MSC26A48
StatusThis copy of the site has no build data. Run m2 status 0WYTM4C9QG.
UsesOne-sided limits of a monotone function exist, Jump discontinuity

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