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In this paper we study the properties of expanding maps with a single discontinuity on a closed interval and the resultant dynamics. For such a map, there exists a compact invariant subset which shares a lot of common properties with classical attractors such as the topological transitivity of the restricted map and the density of the periodic points. The invariant set, with more conditions on the boundary, can be shown to have an isolating neighborhood, hence is a chaotic attractor in the strong sense. Not all such maps derive trapping regions, yet by perturbation, those non-attractors can be made to have a trapping region.