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We show the existence of L\'evy-type stochastic processes in one spacedimension with characteristic triplets that are either discontinuous atthresholds, or are stable-like with stability index functions for which theclosures of the discontinuity sets are countable. For this purpose, weformulate the problem in terms of a Skorokhod-space martingale problemassociated with non-local operators with discontinuous coefficients. Theseoperators are approximated along a sequence of smooth non-local operatorsgiving rise to Feller processes with uniformly controlled symbols. Theyconverge uniformly outside of increasingly smaller neighborhoods of a Lebesguenullset on which the singularities of the limit operator are located.

Solutions of martingale problems for L\'evy-type operators and  stochastic differential equations driven by L\'evy processes with  discontinuous coefficients