  We show that Sturm's classical comparison theorem (SCT) on the interlacing of
zeros of solutions of pairs of real second order two-term ordinary differential
equations necessarily fails if the usual Sturmian-type conditions on the
coefficients are violated. We also show that the conditions on the coefficients
are, in some sense, best possible. We give a necessary and sufficient condition
for the failure of SCT and we note that its converse is false, and that the
comparison theorem may still hold with very mild hypotheses on the
coefficients.
