Editing Absolute continuity (section)

==Absolute continuity of functions==

A continuous function fails to be absolutely continuous if it fails to be [[uniformly continuous]], which can happen if the domain of the function is not compact – examples are tan(''x'') over {{closed-open|0, ''π''/2}}, ''x''<sup>2</sup> over the entire real line, and sin(1/''x'') over (0, 1]. But a continuous function ''f'' can fail to be absolutely continuous even on a compact interval. It may not be "differentiable almost everywhere" (like the [[Weierstrass function]], which is not differentiable anywhere). Or it may be [[Differentiable function|differentiable]] almost everywhere and its derivative ''f''&nbsp;{{prime}} may be [[Lebesgue integration|Lebesgue integrable]], but the integral of ''f''&nbsp;{{prime}} differs from the increment of ''f'' (how much ''f'' changes over an interval). This happens for example with the [[Cantor function]].

===Definition===
Let <math>I</math> be an [[Interval (mathematics)|interval]] in the [[real line]] <math>\R</math>. A function <math>f\colon I \to \R</math> is '''absolutely continuous''' on <math>I</math> if for every positive number <math>\varepsilon</math>, there is a positive number <math>\delta</math> such that whenever a finite sequence of [[pairwise disjoint]] sub-intervals <math>(x_k, y_k)</math> of <math>I</math> with <math>x_k < y_k</math> satisfies<ref>{{harvnb|Royden|1988|loc=Sect. 5.4, page 108}}; {{harvnb|Nielsen|1997|loc=Definition 15.6 on page 251}}; {{harvnb|Athreya|Lahiri|2006|loc=Definitions 4.4.1, 4.4.2 on pages 128,129}}. The interval <math>I</math> is assumed to be bounded and closed in the former two books but not the latter book.</ref>
:<math>\sum_{k=1}^{N} (y_k - x_k) < \delta </math>
then
:<math> \sum_{k=1}^{N} | f(y_k) - f(x_k) | < \varepsilon.</math>

The collection of all absolutely continuous functions on <math>I</math> is denoted <math>\operatorname{AC}(I)</math>.

===Equivalent definitions===

The following conditions on a real-valued function ''f'' on a compact interval [''a'',''b''] are equivalent:<ref>{{harvnb|Nielsen|1997|loc=Theorem 20.8 on page 354}}; also {{harvnb|Royden|1988|loc=Sect. 5.4, page 110}} and {{harvnb|Athreya|Lahiri|2006|loc=Theorems 4.4.1, 4.4.2 on pages 129,130}}.</ref>

# ''f'' is absolutely continuous;
# ''f'' has a derivative ''f''&nbsp;{{prime}} [[almost everywhere]], the derivative is Lebesgue integrable, and <math display="block"> f(x) = f(a) + \int_a^x f'(t) \, dt </math> for all ''x'' on [''a'',''b''];
# there exists a Lebesgue integrable function ''g'' on [''a'',''b''] such that <math display="block"> f(x) = f(a) + \int_a^x g(t) \, dt </math> for all ''x'' in [''a'',''b''].

If these equivalent conditions are satisfied, then necessarily any function ''g'' as in condition 3. satisfies ''g'' = ''f''&nbsp;{{prime}} almost everywhere.

Equivalence between (1) and (3) is known as the '''fundamental theorem of Lebesgue integral calculus''', due to [[Lebesgue]].<ref>{{harvnb|Athreya|Lahiri|2006|loc=before Theorem 4.4.1 on page 129}}.</ref>

For an equivalent definition in terms of measures see the section [[#Relation between the two notions of absolute continuity|Relation between the two notions of absolute continuity]].

===Properties===
* The sum and difference of two absolutely continuous functions are also absolutely continuous. If the two functions are defined on a bounded closed interval, then their product is also absolutely continuous.<ref>{{harvnb |Royden|1988|loc=Problem 5.14(a,b) on page 111}}.</ref>
* If an absolutely continuous function ''f'' is defined on a bounded closed interval and is nowhere zero then ''1/f'' is absolutely continuous.<ref>{{harvnb |Royden|1988|loc=Problem 5.14(c) on page 111}}.</ref>
* Every absolutely continuous function (over a compact interval) is [[uniform continuity|uniformly continuous]] and, therefore, [[Continuous function|continuous]]. Every (globally) [[Lipschitz continuity|Lipschitz-continuous]] [[function (mathematics)|function]] is absolutely continuous.<ref>{{harvnb |Royden|1988|loc=Problem 5.20(a) on page 112}}.</ref>
* If ''f'': [''a'',''b''] → '''R''' is absolutely continuous, then it is of [[bounded variation]] on [''a'',''b''].<ref>{{harvnb|Royden|1988|loc=Lemma 5.11 on page 108}}.</ref>
*  If ''f'': [''a'',''b''] → '''R''' is absolutely continuous, then it can be written as the difference of two monotonic nondecreasing absolutely continuous functions on [''a'',''b''].
* If ''f'': [''a'',''b''] → '''R''' is absolutely continuous, then it has the [[Luzin N property|Luzin ''N'' property]] (that is, for any <math>N \subseteq [a,b]</math> such that <math>\lambda(N) = 0</math>, it holds that <math>\lambda(f(N)) = 0</math>, where <math>\lambda</math> stands for the [[Lebesgue measure]] on '''R''').
* ''f'': ''I'' → '''R''' is absolutely continuous if and only if it is continuous, is of bounded variation and has the Luzin ''N'' property. This statement is also known as the Banach-Zareckiǐ theorem.<ref>{{harvnb |Bruckner|Bruckner|Thomson|1997|loc=Theorem 7.11}}.</ref>
* If ''f'': ''I'' → '''R''' is absolutely continuous and ''g'': '''R''' → '''R''' is globally [[Lipschitz continuity|Lipschitz-continuous]], then the composition ''g <math>\circ</math> f'' is absolutely continuous. Conversely, for every function ''g'' that is not globally Lipschitz continuous there exists an absolutely continuous function ''f'' such that <math>\circ</math> f'' is not absolutely continuous.<ref>{{harvnb |Fichtenholz|1923}}.</ref>

===Examples===
The following functions are uniformly continuous but '''not''' absolutely continuous:
* The [[Cantor function]] on [0, 1] (it is of bounded variation but not absolutely continuous);
* The function:<math display="block"> f(x) = \begin{cases}
0, & \text{if }x =0 \\
x \sin(1/x), & \text{if } x \neq 0
\end{cases} </math> on a finite interval containing the origin.

The following functions are absolutely continuous but not α-Hölder continuous:
* The function ''f''(''x'')&nbsp;=&nbsp;''x<sup>β</sup>'' on [0,&nbsp;''c''], for any {{nowrap|0 < ''β'' < ''α'' < 1}}

The following functions are absolutely continuous and [[Hölder condition|α-Hölder continuous]] but not [[Lipschitz continuity|Lipschitz continuous]]:
* The function ''f''(''x'')&nbsp;=&nbsp;{{radic|''x''}} on [0,&nbsp;''c''], for ''α''&nbsp;≤&nbsp;1/2.

===Generalizations===
Let (''X'', ''d'') be a [[metric space]] and let ''I'' be an [[interval (mathematics)|interval]] in the [[real line]] '''R'''. A function ''f'': ''I'' → ''X'' is '''absolutely continuous''' on ''I'' if for every positive number <math>\varepsilon</math>, there is a positive number <math>\delta</math> such that whenever a finite sequence of [[pairwise disjoint]] sub-intervals [''x''<sub>''k''</sub>, ''y''<sub>''k''</sub>] of ''I'' satisfies:

:<math>\sum_{k} \left| y_k - x_k \right| < \delta</math>

then:

:<math>\sum_{k} d \left( f(y_k), f(x_k) \right) < \varepsilon.</math>

The collection of all absolutely continuous functions from ''I'' into ''X'' is denoted AC(''I''; ''X'').

A further generalization is the space AC<sup>''p''</sup>(''I''; ''X'') of curves ''f'': ''I'' → ''X'' such that:<ref>{{harvnb|Ambrosio|Gigli|Savaré|2005|loc=Definition 1.1.1 on page 23}}</ref>

:<math>d \left( f(s), f(t) \right) \leq \int_s^t m(\tau) \,d\tau \text{ for all } [s, t] \subseteq I</math>

for some ''m'' in the [[Lp space|''L''<sup>''p''</sup> space]] ''L''<sup>''p''</sup>(I).

===Properties of these generalizations===
* Every absolutely continuous function (over a compact interval) is [[uniform continuity|uniformly continuous]] and, therefore, [[Continuous function|continuous]]. Every [[Lipschitz continuity|Lipschitz-continuous]] [[function (mathematics)|function]] is absolutely continuous.
* If ''f'': [''a'',''b''] → ''X'' is absolutely continuous, then it is of [[bounded variation]] on [''a'',''b''].
* For ''f'' ∈ AC<sup>''p''</sup>(''I''; ''X''), the [[metric derivative]] of ''f'' exists for ''λ''-[[almost all]] times in ''I'', and the metric derivative is the smallest ''m'' ∈ ''L''<sup>''p''</sup>(''I''; '''R''') such that:<ref>{{harvnb |Ambrosio|Gigli|Savaré|2005|loc=Theorem 1.1.2 on page 24}}</ref><math display="block">d \left( f(s), f(t) \right) \leq \int_s^t m(\tau) \,d\tau \text{ for all } [s, t] \subseteq I.</math>