  Subgradient and Newton algorithms for nonsmooth optimization require
generalized derivatives to satisfy subtle approximation properties:
conservativity for the former and semismoothness for the latter. Though these
two properties originate in entirely different contexts, we show that in the
semi-algebraic setting they are equivalent. Both properties for a generalized
derivative simply require it to coincide with the standard directional
derivative on the tangent spaces of some partition of the domain into smooth
manifolds. An appealing byproduct is a new short proof that semi-algebraic maps
are semismooth relative to the Clarke Jacobian.
