Motivic Lipschitz continuity

In a general set-up for non-archimedean geometry, we show how local Lipschitz continuity implies piecewise Lipschitz continuity (globally on the whole piece) for definable functions. This is joint work with G. Comte and F. Loeser which generalizes previous work by the same three authors for a fixed p-adic field in [GAFA, 2010] and which fits in a broader program at the interplay of arithmetic and non-archimedean geometry.