  We present a new one parameter family of second derivative discontinuous
solutions to the simplest scale invariant linear ordinary differential
equation. We also point out how the construction could be extended to generate
families of higher derivative discontinuous solutions as well. The
discontinuity can occur only for a subset of even order derivatives, viz.,2nd,
4th, 8th, 16th, ....The solutions are shown to break the discrete parity
(reflection) symmetry of the underlying equation. These results are expected to
gain significance in the contemporary search of a new {\em dynamical principle}
for understanding complex phenomena in Nature.
