A novel problem called satellite downlink scheduling problem (SDSP) under breakpoint resume mode (SDSP-BRM) is studied in our paper. Compared to the traditional SDSP where an imaging data has to be completely downloaded at one time, SDSP-BRM allows the data of an imaging data be broken into a number of pieces which can be downloaded in different playback windows. By analyzing the characteristics of SDSP-BRM, we first propose a mixed integer programming model for its formulation and then prove the NP-hardness of SDSP-BRM. To solve the problem, we design a simple and effective heuristic algorithm (SEHA) where a number of problem-tailored move operators are proposed for local searching. Numerical results on a set of well-designed scenarios demonstrate the efficiency of the proposed algorithm in comparison to the general purpose CPLEX solver. We conduct additional experiments to shed light on the impact of the segmental strategy on the overall performance of the proposed SEHA.