Definition:Absolute Continuity/Real Function

Definition
Let $I \subseteq \R$ be a real interval.

A real function $f: I \to \R$ is said to be absolutely continuous it satisfies the following property:


 * For every $\epsilon > 0$ there exists $\delta > 0$ such that the following property holds:


 * For every finite set of pairwise disjoint closed real intervals $\closedint {a_1} {b_1}, \dotsc, \closedint {a_n} {b_n} \subseteq I$ such that:
 * $\ds \sum_{i \mathop = 1}^n \size {b_i - a_i} < \delta$
 * it holds that:
 * $\ds \sum_{i \mathop = 1}^n \size {\map f {b_i} - \map f {a_i} } < \epsilon$

Also see

 * Definition:Continuous Real Function
 * Definition:Uniformly Continuous Real Function