Math 2.0

statement theorem

A monotone function has countably many discontinuities

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.

Explanations

L1 human

Each discontinuity of an increasing function is a jump up. The jumps take disjoint pieces of the vertical axis, so there can be only countably many.

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To read this node, read these 12 nodes in this order. The foundations come first, and the list ends with this node.

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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
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  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

Sources

  1. Web https://en.wikipedia.org/wiki/Monotonic_function, Some basic applications and results

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