For time-homogeneous stochastic differential equations (SDEs) it is enough to know that the coefficients are Lipschitz to conclude existence and uniqueness of a solution, as well as the existence of a strongly convergent numerical method for its approximation. Here we introduce a notion of piecewise Lipschitz functions and study SDEs with a drift coefficient satisfying only this weaker regularity condition. For these SDEs we can construct a strongly convergent approximation scheme, if the set of discontinuities is a sufficiently smooth hypersurface satisfying the geometrical property of being of positive reach. We then arrive at similar conclusions as in the Lipschitz case. We will see that, although SDEs are in the center of our interest, we will talk surprisingly little about probability theory here.