In continuous simulation, differential equations are solved with numerical algorithms. If the derivatives are not continuous over the simulation period, for example in mechanical systems with friction or in fluid flow with transitions from laminar to turbulent flow and the reverse, differential equation solvers encounter problems. These discontinuities can also appear, for example, when a moving body bounces against a wall. In the past, one has solved this problem by locating zeros of so-called switching functions. These methods do not function properly when switching functions have an even number of zeros within one integration step. Furthermore, these methods encounter problems when the discontinuities are in the neighborhood of model singularities. The algorithm developed by the authors overcomes these problems. In their algorithm, the authors use polynomials to extrapolate the value of the discontinuity function and limit the integration steps in the neighborhood of a model singularity such that the integration step never includes the singularity. The paper concludes with some experiments that show that the new algorithm performs well with slightly more work than the previously developed methods. I found this clear and well-written paper to be interesting.