Criterion for the absolute continuity of curves in metric spaces

TL;DR

This paper establishes a criterion for absolute continuity of curves in metric spaces, showing it is equivalent to the absolute continuity of their composition with any Lipschitz function.

Contribution

It provides a necessary and sufficient condition for absolute continuity of curves in metric spaces based on Lipschitz functions.

Findings

Characterization of absolute continuity via Lipschitz functions

Equivalence condition for curves in metric spaces

Theoretical foundation for analysis in metric spaces

Abstract

It is proved that a parameterized curve in a metric space $X$ is absolutely continuous if and only if its composition with any Lipschitz function on $X$ is absolutely continuous.