## Encyclopedia > Absolutely continuous

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# Absolutely continuous

Definition

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

$\sum (y_1 - x_1) \leq \eta \Rightarrow \sum |f(y_i) - f(x_i)| \leq \varepsilon$

Any Lipschitz continuous[?] function is absolutely continuous.

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

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