Generalized subdi erentials: a Baire categorical approach

We use Baire categorical arguments to construct dramatically pathological locally Lipschitz functions. The origins of this approach can be traced back to Banach and Mazurkiewicz (1931) who independently used similar categorical arguments to show that \almost every continuous real-valued function deened on 0,1] is nowhere diierentiable". As with the results of Banach and Mazurkiewicz, it appears that it is easier to show that almost every function possesses a certain property than to construct a single concrete example. Among the most striking results contained in this paper are: Almost every 1-Lipschitz function deened on a Banach space has a Clarke subdiieren-tial mapping that is identically equal to the dual ball; If fT 1 ; T 2 ; : : : ; T n g is a family of maximal cyclically monotone operators deened on a Banach space X then there exists a real-valued locally Lipschitz function g such that @ 0 g(x) = cofT 1 (x); T 2 (x); : : : ; T n (x)g for each x 2 X; In a separable Banach space each non-empty weak compact convex subset in the dual space is identically equal to the approximate subdiierential mapping of some Lipschitz function and for locally Lipschitz functions deened on separable spaces the notions of strong and weak integrability coincide.