Invariants of the bi-Lipschitz contact equivalence of continuous definable function germs

Abstract: We construct an invariant of the bi-Lipschitz contact equivalence of continuous function germs definable in a polynomially bounded o-minimal structure, such as semialgebraic functions. For a germ $f,$ the invariant is given in terms of the leading coefficients of the asymptotic expansions of $f$ along the connected components of the tangency variety of $f.$