Math 2.0

Real functions Corollary

A function with bounded derivative is Lipschitz

Builds on Mean value theoremLipschitz continuity

Statement

Let II be an interval, let f ⁣:I→ℝf \colon I \to \mathbb{R} be continuous and differentiable at each interior point of II, and let K≥0K \ge 0. If ∣f′(x)∣≤K|f'(x)| \le K for each interior point xx of II, then ff is Lipschitz with constant KK.

Remarks

The mean value theorem gives ∣f(y)−f(x)∣=∣f′(c)∣ ∣y−x∣≤K∣y−x∣|f(y) - f(x)| = |f'(c)| \, |y - x| \le K |y - x|. For example, sin⁡\sin is Lipschitz with constant 11 on ℝ\mathbb{R}. Mathlib states a form for convex sets as Convex.lipschitzOnWith_of_nnnorm_deriv_le.

Reading path

To understand this page from the ground up, read these 24 pages in order. The foundations come first.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionLipschitz continuity, definition, no evidence levelLipschitz continuitydefinitionLocal maximum, definition, no evidence levelLocal maximumdefinitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionLocal minimum, definition, no evidence levelLocal minimumdefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSubsequence, definition, no evidence levelSubsequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionFermat's theorem on stationary points, statement, no evidence levelFermat's theorem on s…statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementEvery sequence has a monotone subsequence, statement, no evidence levelEvery sequence has a …statementBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementBoundedness theorem, statement, no evidence levelBoundedness theoremstatementExtreme value theorem, statement, no evidence levelExtreme value theoremstatementRolle's theorem, statement, no evidence levelRolle's theoremstatementMean value theorem, statement, no evidence levelMean value theoremstatementA function with bounded derivative is Lipschitz, statement, no evidence levelA function with bound…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. Lipschitz continuity definition
  3. Local maximum definition
  4. Sequence definition
  5. Limit of a function definition
  6. Local minimum definition
  7. Real sequence definition
  8. Subsequence definition
  9. Continuity at a point definition
  10. Derivative definition
  11. Bounded sequence definition
  12. Limit of a sequence definition
  13. Monotone sequence definition
  14. Continuous function definition
  15. Fermat's theorem on stationary points statement
  16. Convergent sequence definition
  17. Monotone convergence theorem for sequences statement
  18. Every sequence has a monotone subsequence statement
  19. Bolzano–Weierstrass theorem statement
  20. Boundedness theorem statement
  21. Extreme value theorem statement
  22. Rolle's theorem statement
  23. Mean value theorem statement
  24. A function with bounded derivative is Lipschitz statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Lipschitz_continuity, Examples
About this page
Tag0P64VE79X3
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Kindstatement, corollary
WritingW0 stub
MSC26A16
StatusThis copy of the site has no build data. Run m2 status 0P64VE79X3.
UsesMean value theorem, Lipschitz continuity

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