Piecewise continuity of functions definable 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.