Very slowly varying functions. II

This paper is a sequel to papers by Ash, Erdős and Rubel, on very slowly varying functions, and by Bingham and Ostaszewski, on foundations of regular variation. We show that generalizations of the Ash–Erdős–Rubel approach—imposing growth restrictions on the function h, rather than regularity conditions such as measurability or the Baire property—lead naturally to the main result of regular variation, the Uniform Convergence Theorem.