Definition:Absolute Continuity/Real Function

Definition

Let $I \subseteq \R$ be a real interval.

A real function $f: I \to \R$ is said to be absolutely continuous if and only 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 pairwise disjoint closed real intervals $\closedint {a_1} {b_1}, \dotsc, \closedint {a_n} {b_n} \subseteq I$ such that:

$\ds \sum_{i \mathop = 1}^n \size {b_i - a_i} < \delta$

it holds that:

$\ds \sum_{i \mathop = 1}^n \size {\map f {b_i} - \map f {a_i} } < \epsilon$

Also see

• Definition:Continuous Real Function
• Definition:Uniformly Continuous Real Function

• Results about absolutely continuous real functions can be found here.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): absolutely continuous: 1. (of a real function)
• 2013: Francis Clarke: Functional Analysis, Calculus of Variations and Optimal Control ... (previous) ... (next): $1.1$: Basic Definitions
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): absolutely continuous function
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): absolutely continuous function