We work in the space of transitive, piecewise monotone maps of
a fixed modality m
with the topology of uniform convergence. There is an operator on this
space which assigns
to a map its constant slope model. This operator is discontinuous at
points (maps)
where perturbation can lead to a jump in entropy. Alseda and Misiurewicz
conjectured
that these are the only discontinuity points. We confirm the conjecture
by a technique
of "counting preimages". In particular, the operator is continuous if m≤4.