This paper is devoted to studying function perturbations on the transition matrix and topological structure of a singular Boolean network (SBN) via the semi-tensor product of matrices. First, the algebraic form of an SBN is given, and we discuss how the transition matrix of the SBN changes under function perturbations. Then the local uniqueness of solutions to the SBN is studied, under which the impacts of function perturbations on the topological structure are investigated. Finally, examples are given to show the effectiveness of the obtained results. (C) 2017 Elsevier Ltd. All rights reserved.