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The assumption of sufficiently smooth derivatives underlies
much of the analysis of numerical methods for both ordinary and delay
differential equations. However, derivative discontinuities can arise in
ordinary differential equations and usually do arise in delay differential
equations. In this paper, I review two of the approaches for treating
derivative discontinuities, ``tracking'' using the discontinuity tracking
equations and ``detection and location'' using defect error control. I conclude
that neither approach is ideal when it comes to the treatment of derivative
discontinuities in delay differential algebraic equations.
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