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:

$\displaystyle \sum_{i=1}^n \left \vert {b_i - a_i} \right \vert < \epsilon$

it holds that:

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

See also

• Continuity
• Uniform continuity