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

Abstract

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