On the analyticity of WLUD${}^{\infty }$ functions of one variable and WLUD${}^{\infty }$ functions of several variables in a complete non-Archimedean valued field

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, and whose Hahn group is Archimedean. In this paper, we first review the properties of weakly locally uniformly differentiable (WLUD) functions, $k$ times weakly locally uniformly differentiable (WLUD${}^k$) functions, and WLUD${}^{\infty }$ functions at a point or on an open subset of $\mathcal{N}$. Then we show under what conditions a WLUD${}^{\infty }$ function at a point $x_0\in \mathcal{N}$ is analytic in an interval around $x_0$, that is, it has a convergent Taylor series at any point in that interval. We generalize the concepts of WLUD${}^k$ and WLUD${}^{\infty }$ to functions from ${\mathcal{N}}^n$ to $\mathcal{N}$, for any $n\in \mathbb{N}$. Then we formulate conditions under which a WLUD${}^{\infty }$ function at a point ${\mathbf{x}}_0\in {\mathcal{N}}^n$ is analytic at that point.