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The paper deals with a comprehensive theory of mappings, whose local behaviorcan be described by means of linear subspaces, contained in the graphs of two(primal and dual) generalized derivatives. This class of mappings includes thegraphically Lipschitzian mappings and thus a number of multifunctions,frequently arising in optimization and equilibrium problems. The developedtheory makes use of own generalized derivatives, provides us with some calculusrules and reveals a number of interesting connections. In particular, itenables us to construct a modification of the semismooth* Newton method withimproved convergence properties and to derive a generalization of Clarke'sInverse Function Theorem to multifunctions together with new efficientcharacterizations of strong metric (sub)regularity and tilt stability.

On (local) analysis of multifunctions via subspaces contained in graphs of generalized derivatives