  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
H\"older continuous with some exponent $s\geq 0$ and constant $c \geq 0$, 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.
