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Absolute continuity — Wikipedia (en)

Source: Wikipedia (en)
Original: https://en.wikipedia.org/wiki/Absolute_continuity
License (as archived): CC-BY-SA-4.0
Archived: 2026-11-07T08:32:08.021Z
SHA-256 of the archived text: 6c5311d5e0046f105cf896d776e6d90189e06988c69dc35b6da2ada801bb7a57
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Absolute continuity — Wikipedia (en) — Wikipedia (en), https://en.wikipedia.org/wiki/Absolute_continuity (CC-BY-SA-4.0); archived copy: https://metraipolis.com/documento/dc_e409e141fa0b (Metraipolis archive dc_e409e141fa0b, sha256 6c5311d5e0046f105cf896d776e6d90189e06988c69dc35b6da2ada801bb7a57)

BibTeX

@misc{dce409e141fa0b,
  title = {Absolute continuity — Wikipedia (en)},
  howpublished = {Wikipedia (en)},
  url = {https://en.wikipedia.org/wiki/Absolute_continuity},
  note = {License CC-BY-SA-4.0; archived copy https://metraipolis.com/documento/dc_e409e141fa0b; sha256 6c5311d5e0046f105cf896d776e6d90189e06988c69dc35b6da2ada801bb7a57},
  year = {2026}
}

Archived text

Reproduced under CC-BY-SA-4.0. Attribution: Wikipedia (en), https://en.wikipedia.org/wiki/Absolute_continuity. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In calculus and real analysis, absolute continuity is a regularity property of functions that is stronger than uniform continuity, and hence continuity. The notion of absolute continuity allows one to generalize the fundamental theorem of calculus, originally in the framework of Riemann integration, to Lebesgue integration. For real-valued functions on the real line, two interrelated notions appear: absolute continuity of functions and absolute continuity of measures. These two notions are generalized in different directions. The usual derivative of a function is related to the Radon–Nikodym derivative, or density, of a measure. One has the following chains of inclusions for functions over an interval: Lipschitz continuous ⊆ absolutely continuous ⊆ uniformly continuous ⊆ continuous and, for a bounded interval, absolutely continuous ⊆ bounded variation ⊆ differentiable almost everywhere. Some counterexamples which are continuous but not absolutely continuous: tan ( x ) {\displaystyle \tan(x)} over [0, π/2), and x2 over the entire real line, have non-compact domain and are not uniformly continuous nor of bounded variation; the Weierstrass function over any interval is uniformly continuous, but not differentiable anywhere, and hence not of bounded variation; the Cantor function over x ∈ [ 0 , 1 ] {\displaystyle x\in [0,1]} is uniformly continuous, and is differentiable with f ′ ( x ) = 0 {\displaystyle f'(x)=0} for almost all x, but does not satisfy the Fundamental Theorem: 0 = ∫ 0 1 f ′ ( x ) d x ≠ f ( 1 ) − f ( 0 ) = 1 {\displaystyle 0=\textstyle \int _{0}^{1}f'(x)\,dx\neq f(1)-f(0)=1} .

Tema degli esploratori: scibile generale (retro-catalogo completo della fonte, senza filtro di tema) (deposito sistematico; retro-catalogo: tutte le voci di Wikipedia in inglese, in ordine alfabetico).
Fonte: Wikipedia in inglese, voce «Absolute continuity» — https://en.wikipedia.org/wiki/Absolute_continuity
Wikidata Q332504: «concetto matematico» (dati Wikidata in CC0 1.0) — https://www.wikidata.org/wiki/Q332504
Autori: i contributori di Wikipedia (cronologia: https://en.wikipedia.org/wiki/Absolute_continuity?action=history).
Licenza: Creative Commons Attribuzione-Condividi allo stesso modo 4.0 (CC BY-SA 4.0) — https://creativecommons.org/licenses/by-sa/4.0/. Il riuso richiede attribuzione e stessa licenza.
Estratto dell'introduzione non modificato nel contenuto; revisione 1378727829, consultata il 2026-11-07T08:25:33.598Z tramite l'API ufficiale di Wikipedia. Impronta SHA-256 dell'estratto: b1276bcc8f2b263152778cc22294224c15f5ab1537ca592935ba8715727d8964