Definition:Absolute Continuity
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:
- $\displaystyle \sum_{i \mathop = 1}^n \left \vert {b_i - a_i} \right \vert < \delta$
- it holds that:
- $\displaystyle \sum_{i \mathop = 1}^n \left \vert {f \left({b_i}\right) - f \left({a_i}\right)} \right \vert < \epsilon$
- 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:
