Math 2.0

Real functions Definition

Oscillation of a function

Builds on FunctionSupremumInfimumExtended real number lineContinuity at a pointDirichlet functionLebesgue's criterion for Riemann integrability

Definition

Let D⊆ℝD \subseteq \R, let f ⁣:D→ℝf \colon D \to \R be a function, and let A⊆DA \subseteq D be nonempty. The oscillation of ff on AA is osc⁡f(A)=sup⁡x∈Af(x)−inf⁡x∈Af(x)=sup⁡{∣f(x)−f(y)∣:x,y∈A},\operatorname{osc}_f(A) = \sup_{x \in A} f(x) - \inf_{x \in A} f(x) = \sup \set{\abs{f(x) - f(y)} : x, y \in A}, a value in [0,∞][0, \infty]. For x0∈Dx_0 \in D, the oscillation of ff at x0x_0 is osc⁡f(x0)=inf⁡ε>0osc⁡f(D∩(x0−ε,x0+ε)).\operatorname{osc}_f(x_0) = \inf_{\varepsilon > 0} \operatorname{osc}_f\big(D \cap (x_0 - \varepsilon, x_0 + \varepsilon)\big).

Remarks

The quantity osc⁡f(D∩(x0−ε,x0+ε))\operatorname{osc}_f(D \cap (x_0 - \varepsilon, x_0 + \varepsilon)) decreases as ε\varepsilon decreases, so the infimum is also the limit as ε→0+\varepsilon \to 0^+. The function ff is continuous at x0x_0 if and only if osc⁡f(x0)=0\operatorname{osc}_f(x_0) = 0. For each c>0c > 0, the set {x∈D:osc⁡f(x)≥c}\set{x \in D : \operatorname{osc}_f(x) \ge c} is closed in DD. So the set of discontinuities of ff is a countable union of closed subsets of DD.

Example: let f(x)=x2f(x) = x^2 on ℝ\R and x0=1x_0 = 1. On (1−ε,1+ε)(1 - \varepsilon, 1 + \varepsilon) with ε<1\varepsilon < 1, the values fill ((1−ε)2,(1+ε)2)((1 - \varepsilon)^2, (1 + \varepsilon)^2), a spread of 4ε4\varepsilon. This tends to 00, so osc⁡f(1)=0\operatorname{osc}_f(1) = 0, as continuity predicts.

Non-example of continuity: the Dirichlet function, equal to 11 on ℚ\Q and 00 elsewhere, takes both values on every interval. So its oscillation is 11 at every point, and it is continuous nowhere.

Reading path

To understand this page from the ground up, read these 31 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionExtended real number line, object, no evidence levelExtended real number …objectLimit point of a set, definition, no evidence levelLimit point of a setdefinitionPartition of an interval, definition, no evidence levelPartition of an inter…definitionProposition, concept, no evidence levelPropositionconceptSequence, definition, no evidence levelSequencedefinitionSet bounded above, definition, no evidence levelSet bounded abovedefinitionSet, definition, no evidence levelSetdefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionMesh of a partition, definition, no evidence levelMesh of a partitiondefinitionRiemann sum, definition, no evidence levelRiemann sumdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionInfimum, definition, no evidence levelInfimumdefinitionSupremum, definition, no evidence levelSupremumdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionRiemann integrable function, definition, no evidence levelRiemann integrable fu…definitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionCartesian product, definition, no evidence levelCartesian productdefinitionLebesgue's criterion for Riemann integrability, statement, no evidence levelLebesgue's criterion …statementSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionSequential continuity, definition, no evidence levelSequential continuitydefinitionBinary relation, definition, no evidence levelBinary relationdefinitionDirichlet function, counterexample, no evidence levelDirichlet functioncounterexampleFunction, definition, no evidence levelFunctiondefinitionOscillation of a function, definition, no evidence levelOscillation of a func…definition
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. Element definition
  2. Extended real number line object
  3. Limit point of a set definition
  4. Partition of an interval definition
  5. Proposition concept
  6. Sequence definition
  7. Set bounded above definition
  8. Set definition
  9. Limit of a function definition
  10. Mesh of a partition definition
  11. Riemann sum definition
  12. Implication definition
  13. Predicate definition
  14. Real sequence definition
  15. Infimum definition
  16. Supremum definition
  17. Ordered pair definition
  18. Continuity at a point definition
  19. Riemann integrable function definition
  20. Existential quantifier definition
  21. Universal quantifier definition
  22. Limit of a sequence definition
  23. Cartesian product definition
  24. Lebesgue's criterion for Riemann integrability statement
  25. Subset definition
  26. Uniqueness quantifier definition
  27. Sequential continuity definition
  28. Binary relation definition
  29. Dirichlet function counterexample
  30. Function definition
  31. Oscillation of a function definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Oscillation_(mathematics), Definitions
About this page
TagYMGHNBDZCP
URIm2://YMGHNBDZCP
Kinddefinition
WritingW1 stated
MSC26A15
DefinesOscillation of a function
UsesFunction, Supremum, Infimum, Extended real number line, Continuity at a point, Dirichlet function, Lebesgue's criterion for Riemann integrability
Changed2 November 2026

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