We show that based on ideas of nonarchimedean calculus on Levi Civita fields, these problems can be avoided. It is possible to rigorously decide whether a function is differentiable or not at any given point, and if it is, to determine its derivatives to any order, even if the coding exhibits branch points or non-differentiable pieces. We give details of an implementation of the method and examples for its use for typical pathological problems.
K. Shamseddine, M. Berz, in: "Computational Differentiation: Techniques, Applications, and Tools", M. Berz, C. Bischof, G. Corliss, A. Griewank (Eds.) (1996) 37-52, SIAM
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