Hölder and Lipschitz continuity of functions definable over Henselian rank one valued fields

Abstract

Consider a Henselian rank one valued field $K$ of equicharacteristic zero with the three-sorted language $\mathcal{L}$ of Denef--Pas. Let $f:A\to K$ be a continuous $\mathcal{L}$-definable (with parameters) function on a closed bounded subset $A\subset K^n$. The main purpose is to prove that then $f$ is Hölder continuous with some exponent $s\ge 0$ and constant $c\ge 0$, a fortiori, $f$ is uniformly continuous. Further, if $f$ is locally Lipschitz continuous with a constant $c$, then $f$ is (globally) Lipschitz continuous with possibly some larger constant $d$. Also stated are some problems concerning continuous and Lipschitz continuous functions definable over Henselian valued fields.

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Cited by in corpus (1)

• On functions given by algebraic power series over Henselian valued fields