  It was already known that a p-adic, locally Lipschitz continuous
semi-algebraic function is piecewise Lipschitz continuous, where the pieces can
be taken semi-algebraic. We prove that if the function has locally Lipschitz
constant 1, then it is also piecewise Lipschitz continuous with the same
Lipschitz constant 1. We do this by proving the following fine preparation
results for p-adic semi-algebraic functions in one variable. Any such function
can be well approximated by a monomial with fractional exponent such that
moreover the derivative of the monomial is an approximation of the derivative
of the function. We also prove these results in parametrized versions and in
the subanalytic setting.

Logic;Algebraic Geometry