Absolutely continuous

Definition

A function f is absolutely continuous if, <math> \forall \varepsilon > 0, \exists \eta > 0</math> such that for all list <math>(x_1, y_1, x_2, y_2, \ldots, x_n, y_n)</math> verifying <math>x_1<y_1 \leq x_2<y_2 \leq \ldots \leq x_n < y_n </math>, the following implication holds :

<math> \sum (y_1 - x_1) \leq \eta \Rightarrow \sum |f(y_i) - f(x_i)| \leq \varepsilon </math>

Properties

Any Lipschitz continuous[?] function is absolutely continuous.

Any absolutely continuous function is uniformly continuous and, therefore, continuous.