We introduce a new class of functions called weakly b-continuous which is included in the class of weakly beta-continuous functions due to Popa and Noiri [Popa V, Noiri T. Weakly beta-continuous functions. An Univ Timis Ser Mat-Inform 1994:32(2):83-92]. Some characterizations and several properties concerning weak b-continuity are obtained. Furthermore relationships among strong 0-b-continuity, b-continuity, weak b-continuity and faint b-continuity are investigated. (C) 2008 Elsevier Ltd. All rights reserved.