Geometric and functional analysis vol:20 issue:1 pages:68-87
We prove that a (globally) subanalytic function f : X subset of Q(p)(n) -> Q(p) which is locally Lipschitz continuous with some constant C is piecewise (globally on each piece) Lipschitz continuous with possibly some other constant, where the pieces can be taken to be subanalytic. We also prove the analogous result for a subanalytic family of functions f(y) : X-y subset of Q(p)(n) -> Q(p) depending on p-adic parameters. The statements also hold in a semi-algebraic set-up and also in a finite field extension of Q(p). These results are p-adic analogues of results of K. Kurdyka over the real numbers. To encompass the total disconnectedness of p-adic fields, we need to introduce new methods adapted to the p-adic situation.