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

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 \(f'\) $f^{\prime }$ vanishes at infinity. We show that a necessary and sufficient condition for this is that \(f'\) $f^{\prime }$ 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.