Math 2.0

Real functions Monotone functions, bounded variation, absolute continuity Theorem

Banach–Zarecki theorem

Builds on Absolute continuityBounded variationLuzin N propertyContinuous 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 is absolutely continuous on [a,b][a, b] if and only if the following three conditions hold:

  1. ff is continuous on [a,b][a, b];
  2. ff has bounded variation on [a,b][a, b];
  3. ff has the Luzin N property.

Remarks

An absolutely continuous function is continuous and of bounded variation, and covering a null set by small open intervals shows that it has the N property. For the converse, one shows that if ff is differentiable at every point of a measurable set EE, then the outer measure of f(E)f(E) is at most ∫E∣f′∣\int_E \abs{f'}. The N property handles the null set where f′f' fails to exist, and this controls the changes of ff over families of short intervals. No one condition can be dropped. The Cantor function is continuous and monotone but fails the N property. The function xsin⁡(1/x)x \sin(1/x) on [0,1][0, 1] is continuous with the N property, being smooth away from 00, but it does not have bounded variation. A step function is monotone and maps every set to a finite set, but it is not continuous.

Going deeper

For a student

The theorem explains exactly what goes wrong with the Cantor function and other singular functions: they are continuous and monotone, so only the N property can fail. It is often used to prove absolute continuity of compositions and inverses. For example, a continuous strictly increasing function is absolutely continuous exactly when it maps null sets to null sets.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

Show the path as a graph
Absolute continuity, concept, no evidence levelAbsolute continuityconceptBounded variation, concept, no evidence levelBounded variationconceptLimit point of a set, definition, no evidence levelLimit point of a setdefinitionLuzin N property, concept, no evidence levelLuzin N propertyconceptLimit of a function, definition, no evidence levelLimit of a functiondefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionBanach–Zarecki theorem, statement, no evidence levelBanach–Zarecki theoremstatement
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. Absolute continuity concept
  2. Bounded variation concept
  3. Limit point of a set definition
  4. Luzin N property concept
  5. Limit of a function definition
  6. Continuity at a point definition
  7. Continuous function definition
  8. Banach–Zarecki theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Luzin_N_property
About this page
TagZMEYSAHNX3
URIm2://ZMEYSAHNX3
Kindstatement, theorem
WritingW0 stub
MSC26A46, 26A45
StatusThis copy of the site has no build data. Run m2 status ZMEYSAHNX3.
AboutAbsolute continuity, Luzin N property
UsesAbsolute continuity, Bounded variation, Luzin N property, Continuous function
Changed2 November 2026

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