Asymptotically uniform functions: a single hypothesis which solves two old problems

TL;DR

This paper introduces the concept of asymptotically uniform functions, providing necessary and sufficient conditions for their derivatives to vanish at infinity and for the existence of certain improper integrals, solving two classical problems.

Contribution

It generalizes the notion of asymptotic uniformity to higher derivatives and links it to the behavior of derivatives and improper integrals, unifying two longstanding issues.

Findings

Characterization of when derivatives vanish at infinity

Conditions for the existence of one-sided improper integrals

Broad study of asymptotically uniform functions

Abstract

The asymptotic study of a time-dependent function $f$ as the solution of a differential equation often leads to the question of whether its derivative $\dot f$ vanishes at infinity. We show that a necessary and sufficient condition for this is that $\dot f$ is what may be called "asymptotically uniform". We generalize the result to higher order derivatives. We further show that the same property for $f$ itself is also necessary and sufficient for its one-sided improper integrals to exist. On the way, the article provides a broad study of such asymptotically uniform functions.