Editing Real analysis (section)

====Absolute continuity====
{{Main|Absolute continuity}}
'''Definition.''' Let <math>I\subset\mathbb{R}</math> be an [[interval (mathematics)|interval]] on the [[real line]]. A function <math>f:I \to \mathbb{R}</math> is said to be '''''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_1, y_1), (x_2,y_2),\ldots, (x_n,y_n)</math> of <math>I</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 ''I'' 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>
Absolutely continuous functions are continuous: consider the case ''n'' = 1 in this definition.  The collection of all absolutely continuous functions on ''I'' is denoted AC(''I'').  Absolute continuity is a fundamental concept in the Lebesgue theory of integration, allowing the formulation of a generalized version of the fundamental theorem of calculus that applies to the Lebesgue integral.