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Very slowly varying functions - II

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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-Erdös-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. Keywords: Slow variation, Uniform Convergence Theorem, Heiberg-Lipschitz condition, Heiberg-Seneta theorem.

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