# Fine and Pathological Properties of Subdifferentials

Wang, Shawn Xianfu (1999) Fine and Pathological Properties of Subdifferentials. PhD Thesis thesis, Simon Fraser University.

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In this thesis, we explore a Baire categorical approach for studying the subdifferentiability of Lipschitz functions and continuous functions. On the line, we show that nowhere monotone functions are the key ingredients to construction of continuous functions, absolutely continuous functions, and Lipschitz functions with large subdifferentials. We also explicitly construct Lipschitz functions which are simultaneously H\"older subdifferentiable and superdifferentiable only on a countable dense subset. On separable Banach spaces, we prove that the class of functions with maximal Clarke and approximate subdifferentials is residual in an appropriate complete metric space. In particular, in the space of nonexpansive functions endowed with the supremum metric, the set of functions whose Clarke subdifferential and approximate subdifferential are identically equal to the dual unit ball is residual; in the space of bounded continuous functions endowed with the supremum metric, the set of functions whose Clarke subdifferential and approximate subdifferential are identically equal to the whole dual space is residual. On general Banach spaces, we show that for a locally Lipschitz function $f$ there exists a residual subset $G$, in an appropriate complete metric space, such that $(-\partial_{\sharp}(-f))\cap\partial_{\sharp}g\neq\emptyset$ everywhere for each $g\in G$ where $\partial_{\sharp}$ stands for the approximate or Clarke subdifferential. Diverse implications are given. In order to recover a locally Lipschitz function from its Clarke subdifferential, we also study line integrals in general Banach spaces. Such a categorical approach allows us to generalize and simplify many known results on subdifferential integrability, and to illustrate the stark difference between the subdifferentiability of convex functions and the subdifferentiability of nonconvex functions.