  In this paper we describe the Cuntz semigroup of continuous fields of
C$^*$-algebras over one dimensional spaces whose fibers have stable rank one
and trivial $K_1$ for each closed, two-sided ideal. This is done in terms of
the semigroup of global sections on a certain topological space built out of
the Cuntz semigroups of the fibers of the continuous field. When the fibers
have furthermore real rank zero, and taking into account the action of the
space, our description yields that the Cuntz semigroup is a classifying
invariant if and only if so is the sheaf induced by the Murray-von Neumann
semigroup.
