Tame functions are semismooth

Superlinear convergence of the Newton method for nonsmooth equations requires a “semismoothness” assumption. In this work we prove that locally Lipschitz functions definable in an o-minimal structure (in particular semialgebraic or globally subanalytic functions) are semismooth. Semialgebraic, or more generally, globally subanalytic mappings present the special interest of being γ -order semismooth, where γ is a positive parameter. As an application of this new estimate, we prove that the error at the kth step of the Newton method behaves like O(2−(1+γ )k ).