Piecewise continuity of functions definable over Henselian rank one valued fields

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 $A_1,\dots ,A_s$ of $K^n$ such that the restriction of $f$ to each $A_i$ is a continuous function.