This paper is a sequel to both Ash, Erd1⁄2os and Rubel [AER], on very slowly varying functions, and [BOst1], on foundations of regular variation. We show that generalizations of the Ash-Erd1⁄2os-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, th...

We prove two new effective properties of rational functions over infinite words which are realized by finite state Büchi transducers. Firstly, for each such function [Formula: see text], one can construct a deterministic automaton text] accepting dense text]-subset that the restriction to is continuous. Secondly, we give proof decidability first Baire class synchronous text]-rational from get a...

For a subset A of an -group B, r(A,B) denotes the relative uniform closure of A in B. RX denotes the -group of all real-valued functions on the set X, and when X is a topological space, C∗(X) is the -group of all bounded continuous real-valued functions, and B(X) is the -group of all Baire functions. We show that B(X) = r (C∗(X), B(X)) = r ¡ C∗(X), R ¢ . This would appear to be a purely order-t...

We generalize the Baire Category Theorem to the Borel and difference hierarchies, i.e. if Γ is any of the classes Σξ , Π 0 ξ , Dη(Σ 0 ξ) or Ďη(Σ 0 ξ) we find a representative set PΓ ∈ Γ and a Polish topology τΓ such that for every A ∈ Γ̌ from some assumption on the size of A ∩ PΓ we can deduce that A \ PΓ is of second category in the topology τΓ. This allows us to distinguish the levels of the B...

Say that a cardinal number κ is small relative to the space X if κ < ∆(X), where ∆(X) is the least cardinality of a non-empty open set in X . We prove that no Baire metric space can be covered by a small number of discrete sets, and give some generalizations. We show a ZFC example of a regular Baire σ-space and a consistent example of a normal Baire Moore space which can be covered by a small n...

In the theory of optimization an essential role is played by the differentiability of convex functions. In this paper we shall try to extend the results concerning differentiability to a larger class of functions called strongly α(·)-paraconvex. Let (X, ‖.‖) be a real Banach space. Let f(x) be a real valued strongly α(·)-paraconvex function defined on an open convex subset Ω ⊂ X , i.e. f ( tx+(...

 The category of the title is called $mathcal{W}$. This has all free objects $F(I)$ ($I$ a set). For an object class $mathcal{A}$, $Hmathcal{A}$ consists of all homomorphic images of $mathcal{A}$-objects. This note continues the study of the $H$-closed monoreflections $(mathcal{R}, r)$ (meaning $Hmathcal{R} = mathcal{R}$), about which we show ({em inter alia}): $A in mathcal{A}$ if and  only if...

It is known that a real valued measure (1) on the a-ring of Baire sets of a locally compact Hausdorff space, or (2) on the Borel sets of a complete separable metric space is regular. Recently Dinculeanu and Kluvanek used regularity of non-negative Baire measures to prove that any Baire measure with values in a locally convex Hausdorff topological vector space (TVS) is regular. Subsequently a di...