Abstract

Let \(\mathcal{N}\) be a non-Archimedean ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. In this paper, we first review the properties of weakly locally uniformly differentiable (WLUD) functions [1] at a point or on an open subset of \(\mathcal{N}\). WLUD functions are \(C^1\) and they form an \( \mathcal{N} \)-algebra that is closed under composition and contains all polynomial functions. Moreover, they satisfy an inverse function theorem, a local intermediate value theorem and a local mean value theorem. We define \(k\) times weakly locally uniformly differentiable (WLUD\(^k\)) functions from \(\mathcal{N}\) to \(\mathcal{N}\), then we state and prove a Taylor theorem with remainder for WLUD\(^k\) functions on \(\mathcal{N}\). Finally, we generalize the concept of weak local uniform differentiability to functions from \(\mathcal{N}^n\) to \(\mathcal{N}^m\) with \(m,n\in\mathbb{N}\), then we formulate and prove the inverse function theorem for WLUD functions from \(\mathcal{N}^n\) to \(\mathcal{N}^n\) and the implicit function theorem for WLUD functions from \(\mathcal{N}^n\) to \(\mathcal{N}^m\) with \(m<n\) in \(\mathbb{N}\).

中文翻译：

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非阿奇米德空间上的弱局部一致可分函数的泰勒定理，逆函数定理和隐函数定理$$ ^ * $$

摘要

令\（\ mathcal {N} \）为实数的非阿希米德有序字段扩展，该实扩展是由该订单诱导的拓扑中的实封闭且柯西完整的。在本文中，我们首先回顾\（\ mathcal {N} \）的一个点或一个开放子集上的弱局部一致可微（WLUD）函数的性质[1] 。WLUD函数为\（C ^ 1 \），它们形成一个\（\ mathcal {N} \）-代数，该代数在合成下关闭并包含所有多项式函数。此外，它们满足逆函数定理，局部中间值定理和局部平均值定理。我们定义\（k \）次的弱局部一致可分（WLUD \（^ k \））函数\（\ mathcal {N} \）到\（\ mathcal {N} \），然后陈述并证明\（\ mathcal {N} \）上WLUD \（^ k \）函数具有余数的泰勒定理。最后，我们将弱局部一致微分的概念推广到\（\ mathcal {N} ^ n \）到\（\ mathcal {N} ^ m \）和\（m，n \ in \ mathbb {N} \） ，那么我们制定并证明从WLUD功能的逆功能定理\（\ mathcal {N} ^ N \）到\（\ mathcal {N} ^ N \） ，并从用于WLUD功能隐函数定理\（ \ mathcal {N} ^ N \）到\（\ mathcal {N} ^ M \）与\（M <N \）在\（\ mathbb {N} \）。