Editing Absolute continuity (section)

===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>