Integrable functions with no limit at infinity

One of the classical topics in analysis concerns functions that are integrable on an unbounded interval but do not tend to zero at infinity. In this paper, we establish general convergence and divergence criteria for the integrals of a class of such non-vanishing functions introduced by du Bois-Reymond and Gilbert, and subsequently studied by Hardy. We establish a more general convergence criterion that extends Gilbert’s result, which is a special case of our theorem.