Int. J. Contemp. Math. Sciences, Vol. 3, 2008, no. 30, 1469 - 1480 On Almost b-Continuous Functions U˘ gur S ¸eng¨ ul Department of Mathematics, Faculty of Science and Letters Marmara University, 34722 G¨oztepe- ˙ Istanbul, Turkey usengul@marmara.edu.tr Abstract In this paper we introduce a new class of functions called almost b-continuous. Some characterizations and several properties concerning almost b-continuity are obtained. Mathematics Subject Classification: 54C08, 54C10 Keywords: b-open sets, almost b-continuous functions, weakly b-continuous functions, b-strongly closed graph 1 Introduction b-open sets is introduced by Andrijevic [3] and variants of continuity related to this concept, studied in [6,11, 12, 21, 23]. The purpose of the present paper is to introduce and investigate some of the fundamental properties of almost b- continuous functions. We obtain characterizations of almost b-continuous func- tions and investigate relationships between almost b-continuity and some other variants of b-continuity. In Section 3 basic properties of almost b-continuous functions are given. In Section 4, we investigate relationships between almost b-continuity and separation axioms and study relationships between almost b-continuity and b-strongly closed graphs. 2 Preliminaries Throughout this paper (X, τ ) and (Y,σ) (or simply X and Y ) represent nonempty topological spaces on which no separation axioms are assumed, unless other- wise mentioned. For a subset S of (X, τ ), cl(S ) and int(S ) represent the closure of S with respect to τ and the interior of S with respect to τ , re- spectively. A subset S of a space (X, τ ) is called regular open [20] (resp.