  We aim to study a one-dimensional $p$-wave superconductor with quasiperiodic
on-site potentials. A modified real-space-Pfaffian method is applied to
calculate the topological invariants. We confirm that the Majorana zero mode is
protected by the nontrivial topology the topological phase transition is
accompanied by the energy gap closing and reopening. In addition, we
numerically find that there are mobility edges which originate from the
competition between the extended $p$-wave pairing and the localized
quasi-disorder. We qualitatively analyze the influence of superconducting
pairing parameters and on-site potential strength on the mobility edge. In
general, our work enriches the research on the $p$-wave superconducting models
with quasiperiodic potentials.
