  It was pointed out that the black hole quasinormal modes resulting from a
piecewise approximate potential are drastically distinct from those pertaining
to the original black hole metric. In particular, instead of lining up parallel
to the imaginary axis, the spectrum is found to stretch out along the real
axis. In this work, we prove that if there is a single discontinuity in the
effective potential, no matter how insignificant it is, the asymptotical
behavior of the quasinormal modes will be appreciably modified. Besides showing
numerical evidence, we give analytical derivations to support the above
assertion even when the discontinuity is located significantly further away
from the maximum of the potential and/or the size of the step is arbitrarily
small. Moreover, we discuss the astrophysical significance of the potential
implications in terms of the present findings.
