TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 353, Number 10, Pages 3875–3893 S 0002-9947(01)02820-3 Article electronically published on May 14, 2001 GENERALIZED SUBDIFFERENTIALS: A BAIRE CATEGORICAL APPROACH JONATHAN M. BORWEIN, WARREN B. MOORS, AND XIANFU WANG Abstract. We use Baire categorical arguments to construct pathological lo- cally Lipschitz functions. The origins of this approach can be traced back to Banach and Mazurkiewicz (1931) who independently used similar categor- ical 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 func- tion possesses a certain property than to construct a single concrete exam- ple. Among the most striking results contained in this paper are: Almost every 1-Lipschitz function defined on a Banach space has a Clarke subdiffer- ential mapping that is identically equal to the dual ball; if {T 1 ,T 2 ,...,Tn} 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 ∂ 0 g(x) = co{T 1 (x),T 2 (x),... ,Tn(x)} for each x ∈ X; in a separable Banach space each non-empty weak ∗ compact convex subset in the dual space is iden- tically 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. 1. Introduction An important aspect of developing a mathematical theory is in producing both examples and counterexamples that illuminate the content and boundaries of the subject. In this paper we give a general method for constructing examples and counterexamples for the differentiability theory of Lipschitz functions. The first and perhaps best known counterexample in differentiability theory is the construction of a continuous nowhere differentiable function. The explicit con- structions given in the 19th century were later (in 1931) augmented by the use of Baire categorical arguments. Since this time the use of Baire category for the construction of functions (either well behaved or pathological) has been applied to several areas of analysis, (see, [12], [13] and [23] to name but a few). In this paper we continue this tradition by using Baire category arguments to construct Lipschitz functions that have ‘large’ generalized derivatives. The first and most crucial step towards achieving this result is to produce a candidate complete metric space on Received by the editors March 24, 1999 and, in revised form, February 25, 2000. 1991 Mathematics Subject Classification. Primary 49J52, 54E52. Key words and phrases. Subdifferentials, differentiability, Baire category, upper semi– continuous set–valued map, T -Lipschitz function. Research of the first author was supported by NSERC and the Shrum endowment of Simon Fraser University. Research of the second author was supported by a Marsden fund grant, VUW 703, administered by the Royal Society of New Zealand. c 2001 American Mathematical Society 3875 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use