Generalized subdifferentials: a Baire categorical approach

Abstract:

We use Baire categorical arguments to construct 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 defined on [0,1] is nowhere differentiable". 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 defined on a Banach space has a Clarke subdifferential mapping that is identically equal to the dual ball; if $\{T_{1}, T_{2},\ldots , T_{n}\}$ is a family of maximal cyclically monotone operators defined on a Banach space $X$ then there exists a real-valued locally Lipschitz function $g$ such that $\partial _{0}g(x)=\mbox {co}\{T_{1}(x),T_{2}(x),\ldots , T_{n}(x)\}$ for each $x\in X$; in a separable Banach space each non-empty weak$^{*}$ compact convex subset in the dual space is identically equal to the approximate subdifferential mapping of some Lipschitz function and for locally Lipschitz functions defined on separable spaces the notions of strong and weak integrability coincide.