In this paper, we introduce a new class of functions called weakly BR-continuous function including the classes of strongly theta-b-continuous functions due to Park [Park JH. Strongly theta-b-continuous functions. Acta Math Hungar 2006; 110(4):347-59] and weakly clopen functions due to Son et al. [Son MJ, Park JH, Lim KM. Weakly clopen functions. Chaos, Solitons & Fractals 2007;33:1746-55]. Characterizations and properties of the class of weakly BR-continuous functions are investigated. Moreover, the relationships among weakly BR-continuous functions, strongly theta-b-continuous functions, weakly clopen functions and the related functions are investigated. (c) 2008 Elsevier Ltd. All rights reserved.