---
tag: ZT6F4C2YHK
kind: concept
name: Absolute continuity
msc: [26A46]
sources:
  - url: https://en.wikipedia.org/wiki/Absolute_continuity
same_as:
  wikidata: Q332504
  wikipedia: Absolute continuity
---

## Summary

A real function on a closed interval is absolutely continuous when small total length of input intervals forces small total change of the output.
It is stronger than [[uniform continuity|0Q4DR37DVE]] and stronger than [[bounded variation|A8251BHCMP]].
The absolutely continuous functions are exactly the indefinite [[Lebesgue integrals|DWJF7ABPR1]].
So they are the functions for which the [[fundamental theorem of calculus|RX7JSHMP8Z]] holds in Lebesgue's form.
The analogous notion for measures is the [[absolutely continuous measure|P0P21KDV6E]].

## Explanations

### L1 human

Uniform continuity says that one short input interval gives a small change of the output.
Absolute continuity asks the same for many short intervals at once, as long as their total length is small.
Every [[Lipschitz function|0QNVQ4VD9T]], such as $\sin x$, is absolutely continuous.
The [[Cantor function|NPBYYK63FE]] is continuous and increasing but not absolutely continuous: it rises from $0$ to $1$ on a set of total length $0$.