Abstract
We introduce a class of so-called very weakly locally uniformly differentiable (VWLUD) functions at a point of a general non-Archimedean ordered field extension of the real numbers, \(\mathcal{N}\), which is real closed and Cauchy complete in the topology induced by the order, and whose Hahn group is Archimedean. This new class of functions is defined by a significantly weaker criterion than that of the class of weakly locally uniformly differentiable (WLUD) functions studied in [1], which is nonetheless sufficient for a slight variation of the inverse function theorem and intermediate value theorem. Similarly, a weaker second order criterion is derived from the previously studied WLUD\(^2\) condition for twice-differentiable functions. We show that VWLUD\(^2\,\) functions at a point of \(\mathcal{N}\) satisfy the mean value theorem in an interval around that point.
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Shamseddine, K., Shalev, A. A Weaker Smoothness Criterion for the Inverse Function Theorem, the Intermediate Value Theorem, and the Mean Value Theorem in a non-Archimedean Setting. P-Adic Num Ultrametr Anal Appl 14 (Suppl 1), S45–S58 (2022). https://doi.org/10.1134/S2070046622050041
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DOI: https://doi.org/10.1134/S2070046622050041