Math 2.0

Real functions Monotone functions, bounded variation, absolute continuity Theorem

Jordan decomposition of a function of bounded variation

Builds on Bounded variationTotal variationMonotone function

Statement

Let a<ba < b be real numbers and let f ⁣:[a,b]→ℝf \colon [a, b] \to \R be a function. Then ff has bounded variation on [a,b][a, b] if and only if there are nondecreasing functions g,h ⁣:[a,b]→ℝg, h \colon [a, b] \to \R with f=g−hf = g - h. Moreover, when ff has bounded variation, one can take g(x)=Vax(f)g(x) = V_a^x(f) and h(x)=Vax(f)−f(x)h(x) = V_a^x(f) - f(x), where Vax(f)V_a^x(f) is the total variation of ff on [a,x][a, x] and Vaa(f)=0V_a^a(f) = 0.

Remarks

A nondecreasing function has bounded variation, and a difference of two functions of bounded variation has bounded variation, which gives one direction. For the other direction, g(x)=Vax(f)g(x) = V_a^x(f) is nondecreasing because variation is additive over adjacent intervals. For x<yx < y one has f(y)−f(x)≤∣f(y)−f(x)∣≤Vxy(f)=g(y)−g(x)f(y) - f(x) \le \abs{f(y) - f(x)} \le V_x^y(f) = g(y) - g(x), so h=g−fh = g - f is nondecreasing too. For example, f(x)=sin⁡xf(x) = \sin x on [0,2π][0, 2\pi] is the difference of g(x)=V0x(sin⁡)g(x) = V_0^x(\sin) and g−sin⁡g - \sin. The decomposition is not unique: one may add the same nondecreasing function to gg and to hh.

Going deeper

For a student

With the decomposition, results about monotone functions pass to functions of bounded variation. By Lebesgue's theorem on monotone functions, a function of bounded variation is differentiable almost everywhere. It also has one-sided limits at every point and at most countably many discontinuities. The theorem is the one-variable model of the Jordan decomposition of a signed measure.

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Bounded variation, concept, no evidence levelBounded variationconceptElement, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptTotal variation, concept, no evidence levelTotal variationconceptSet, definition, no evidence levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionFunction, definition, no evidence levelFunctiondefinitionPreorder, definition, no evidence levelPreorderdefinitionMonotone function, definition, no evidence levelMonotone functiondefinitionJordan decomposition of a function of bounded variation, statement, no evidence levelJordan decomposition …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. Bounded variation concept
  2. Element definition
  3. Proposition concept
  4. Total variation concept
  5. Set definition
  6. Implication definition
  7. Predicate definition
  8. Ordered pair definition
  9. Existential quantifier definition
  10. Universal quantifier definition
  11. Cartesian product definition
  12. Subset definition
  13. Uniqueness quantifier definition
  14. Binary relation definition
  15. Reflexive relation definition
  16. Transitive relation definition
  17. Function definition
  18. Preorder definition
  19. Monotone function definition
  20. Jordan decomposition of a function of bounded variation statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Bounded_variation
About this page
TagKDT3E0VE0K
URIm2://KDT3E0VE0K
Kindstatement, theorem
WritingW0 stub
MSC26A45, 26A48
StatusThis copy of the site has no build data. Run m2 status KDT3E0VE0K.
AboutBounded variation
UsesBounded variation, Total variation, Monotone function
Changed2 November 2026

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