When solving delay differential equations (DDEs) with state-dependent delays the problem of breaking point detection is important. Points where the solution is not smooth enough to provide the order of the method must be included into the computational mesh, otherwise a reduction in the order of the solution will result. The problem, however is to detect and compute such points efficiently. Breaking points arise every time a delay falls on a previous breaking point (either of the calculated solution or in the history function). In the case of retarded DDEs the new breaking point is (at least) one order smoother than the previous breaking point that gave rise to it. For fixed or time-dependent delays the breaking points can be precomputed independent of the solution, but for state-dependent delays the positions of the breaking points depend on the computed solution. If a breaking point is detected and the step-size is changed in order to incorporate the point into the mesh, then the new step-size generates a n