Piecewise continuity of functions definable over Henselian rank one valued fields

TL;DR

This paper proves that functions definable over Henselian rank one valued fields of equicharacteristic zero are piecewise continuous, with the domain partitioned into finitely many definable locally closed subsets where the function is continuous.

Contribution

It establishes a piecewise continuity property for definable functions over Henselian rank one valued fields in the Denef--Pas language, extending understanding of definable functions in valued field model theory.

Findings

Definable functions are piecewise continuous on finitely many definable locally closed subsets.

The domain can be partitioned into finitely many parts where the function is continuous.

The result applies to functions definable in the Denef--Pas language over Henselian rank one valued fields.

Abstract

Consider a Henselian rank one valued field $K$ of equicharacteristic zero along with the language $\mathcal{L}^{P}$ of Denef--Pas. Let $f: A \to K$ be an $\mathcal{L}^{P}$-definable (with parameters) function on a subset $A$ of $K^{n}$. We prove that $f$ is piecewise continuous; more precisely, there is a finite partition of $A$ into $\mathcal{L}^{P}$-definable locally closed subsets $A_{1},\ldots,A_{s}$ of $K^{n}$ such that the restriction of $f$ to each $A_{i}$ is a continuous function.