  By a Morse function on a compact manifold with boundary we mean a real-valued
function without critical points near the boundary such that its critical
points as well as the critical points of its restriction to the boundary are
all non-degenerate. For such Morse functions, Saeki and Yamamoto have
previously defined a certain notion of cusp cobordism, and computed the
unoriented cusp cobordism group of Morse functions on surfaces. In this paper,
we compute unoriented and oriented cusp cobordism groups of Morse functions on
manifolds of any dimension by employing Levine's cusp elimination technique as
well as the complementary process of creating pairs of cusps along fold lines.
We show that both groups are cyclic of order two in even dimensions, and cyclic
of infinite order in odd dimensions. For Morse functions on surfaces our result
yields an explicit description of Saeki-Yamamoto's cobordism invariant which
they constructed by means of the cohomology of the universal complex of
singular fibers.
