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There has been substantial interest in developing Markov chain Monte Carloalgorithms based on piecewise-deterministic Markov processes. However existingalgorithms can only be used if the target distribution of interest isdifferentiable everywhere. The key to adapting these algorithms so that theycan sample from to densities with discontinuities is defining appropriatedynamics for the process when it hits a discontinuity. We present a simplecondition for the transition of the process at a discontinuity which can beused to extend any existing sampler for smooth densities, and give specificchoices for this transition which work with popular algorithms such as theBouncy Particle Sampler, the Coordinate Sampler and the Zig-Zag Process. Ourtheoretical results extend and make rigorous arguments that have been presentedpreviously, for instance constructing samplers for continuous densitiesrestricted to a bounded domain, and we present a version of the Zig-Zag Processthat can work in such a scenario. Our novel approach to deriving the invariantdistribution of a piecewise-deterministic Markov process with boundaries may beof independent interest.

PDMP Monte Carlo methods for piecewise-smooth densities