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 if it satisfies the following property:


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


 * For every finite set of disjoint closed real intervals $\left[{a_1 . . b_1}\right], \ldots, \left[{a_n . . b_n}\right] \subseteq I$ such that:
 * $\displaystyle \sum_{i=1}^n \left \vert {b_i - a_i} \right \vert < \epsilon$
 * it holds that:
 * $\displaystyle \sum_{i=1}^n \left \vert {f \left({b_i}\right) - f \left({a_i}\right)} \right \vert < \delta$