Definition:Absolute Continuity

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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:
$\displaystyle \sum_{i \mathop = 1}^n \left \vert {f \left({b_i}\right) - f \left({a_i}\right)} \right \vert < \epsilon$


Also see