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

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.