Border-collision bifurcations of a class of one-dimensional piecewise discontinuous maps were investigated. The border-collision bifurcation curves and fold bifurcation conditions for period-n solutions were derived. The correctness of the deduced results was confirmed through the numerical simulations. It is found that there are period adding sequences and period overlapping sequences in the system. Finally, the three-parameter bifurcations of the piecewise discontinuous map were considered, which demonstrates the combined influence of all the parameters in the map.