  We show the existence of L\'evy-type stochastic processes in one space
dimension with characteristic triplets that are either discontinuous at
thresholds, or are stable-like with stability index functions for which the
closures of the discontinuity sets are countable. For this purpose, we
formulate the problem in terms of a Skorokhod-space martingale problem
associated with non-local operators with discontinuous coefficients. These
operators are approximated along a sequence of smooth non-local operators
giving rise to Feller processes with uniformly controlled symbols. They
converge uniformly outside of increasingly smaller neighborhoods of a Lebesgue
nullset on which the singularities of the limit operator are located.
