Discontinuous solutions of neutral delay differential equations

It is well known that the solutions of delay differential and implicit and explicit neutral delay differential equations (NDDEs) may have discontinuous derivatives, but it has not been appreciated (sufficiently) that the solutions of NDDEs—and, therefore, solutions of delay differential algebraic equations—need not be continuous. Numerical codes for solving differential equations, with or without retarded arguments, are generally based on the assumption that a solution is continuous. We illustrate and explain how the discontinuities arise, and present some methods to deal with these problems computationally. The investigation of a simple example is followed by a discussion of more general NDDEs and further mathematical detail.