  We prove a nonsmooth implicit function theorem applicable to the zero set of
the difference of convex functions. This theorem is explicit and global: it
gives a formula representing this zero set as a difference of convex functions
which holds throughout the entire domain of the original functions. As
applications, we prove results on the stability of singularities of envelopes
of semi-convex functions, and solutions to optimal transport problems under
appropriate perturbations, along with global structure theorems on certain
discontinuities arising in optimal transport maps for Ma-Trudinger-Wang costs.
For targets whose components satisfy additional convexity, separation,
multiplicity and affine independence assumptions we show these discontinuities
occur on submanifolds of the appropriate codimension which are parameterized
locally as differences of convex functions (DC, hence $C^2$ rectifiable), and
--- depending on the precise assumptions --- $C^{1,\alpha}$ smooth. In this
case the highest codimension submanifolds consists of isolated points, each
uniquely identified by the (affinely independent) components of the target to
which it is transported.
