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.

中文翻译：

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非阿基米德设置中反函数定理、中值定理和中值定理的较弱平滑准则

摘要

我们在实数的一般非阿基米德有序域扩展\(\mathcal{N}\) 的一点上引入了一类所谓的非常弱的局部一致可微 (VWLUD) 函数，它是实闭且Cauchy在由阶诱导的拓扑中完成，并且其哈恩群是阿基米德。这一类新函数的定义标准明显弱于 [1] 中研究的弱局部一致可微 (WLUD) 函数类标准，但足以对反函数定理和中间值定理进行轻微变化。类似地，较弱的二阶准则是从先前研究的二次可微函数的WLUD \(^2\)条件导出的。我们证明 VWLUD \(^2\,\)在\(\mathcal{N}\)点的函数在该点周围的区间内满足中值定理。