  The properties of the momentum space quark propagator in Landau gauge are
examined for the overlap quark action in quenched lattice QCD. Numerical
calculations are done on three lattices with different lattice spacings and
similar physical volumes to explore the approach of the quark propagator
towards the continuum limit. We have calculated the nonperturbative
momentum-dependent wavefunction renormalization function $Z(p^2)$ and the
nonperturbative mass function $M(p^2)$ for a variety of bare quark masses and
extrapolate to the chiral limit.
  We find the behavior of $Z(p^2)$ and $M(p^2)$ are in good agreement for the
two finer lattices in the chiral limit. The quark condensate is also
calculated.
