VERY SLOWLY VARYING FUNCTIONS – II BY

Abstract. This paper is a sequel to both Ash, Erdös and Rubel [AER], on very slowly varying functions, and [BOst1], 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 mea-surability or the Baire property – lead naturally to the main result of regular variation, the Uniform Convergence Theorem. 1 Introduction and Main Result We work with the Karamata theory of regular and slow variation; see [BGT] – BGT in what follows – for a monograph account. Here the main result is the Uniform Convergence Theorem – UCT below – which asserts that the defining pointwise convergence for slow variation in fact holds uniformly on compac