We explain in this paper why continuous coarse spaces are a suboptimal choice for domain decomposition methods that have discontinuous iterates, like for example restricted Additive Schwarz methods, or optimized Schwarz methods. As an alternative, we propose discontinuous coarse spaces for such methods. For linear problems, we show how to design one such discontinuous coarse space and present an algorithm that computes an efficient discontinuous coarse space correction for the special case of an optimized Schwarz method. While the algorithm is suitable for higher dimensions, it has the special property of converging in a single coarse iteration for one-dimensional linear problems. We illustrate our new algorithm by numerical experiments.