  For the numerical solution of nonsmooth problems, sometimes it is not
necessary that an exact subgradient/generalized Jacobian is at our disposal,
but that a certain semismoothness property is fulfilled. In this paper we
consider not only semismoothness of nonsmooth real- and vector-valued mappings,
but also its interplay with the semismoothness$^*$ property for multifunctions.
In particular, we are interested in the semismoothness of solution maps to
parametric semismooth$^*$ inclusions. Our results are expressed in terms of
suitable generalized derivatives of the set-valued part, i.e., by limiting
coderivatives or by SC (subspace containing) derivatives. As a byproduct we
identify a class of multifunctions having the remarkable property that they are
strictly proto-differentiable almost everywhere (with respect to some Hausdorff
measure) on their graph.
