The stability of a high-symmetry $P4mm$ polar phase in seventeen short-period ferroelectric perovskite superlattices with polar discontinuity is studied from first principles within the density-functional theory. It is shown that in most superlattices this phase exhibits either the ferroelectric instability or the antiferrodistortive one, or both of them. For each superlattice, the ground-state structure, the structure of possible metastable phases, the spontaneous polarization, and piezoelectric properties are calculated. The properties of superlattices with the polar discontinuity are compared with those of superlattices with broken symmetry and of ordinary superlattices, which have no the polar discontinuity. It is shown that high piezoelectric coefficients (up to 150--270 pC/N) in some superlattices with the polar discontinuity are due to the appearance of strong lattice distortions, whose symmetry follows that of a low-lying polar phonon mode of the ground-state structure, under the influence of external strain.