Definition:Absolute Continuity/Real Function

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

it holds that
 * $$\sum_{i=1}^n \left \vert {f \left({b_i}\right) - f \left({a_i}\right)} \right \vert < \delta$$.