Subdifferentiability of Real Functions

Abstract

In this paper, we show that nowhere monotone functions are the key ingredients to construction of continuous functions, absolutely continuous functions, and Lipschitz functions with large subdifferentials on the real line. Let $\partial_{c}f, \partial_{a}f$ denote the Clarke subdifferential and approximate subdifferential respectively. We construct absolutely continuous functions on $\R$ such that $\partial_{a}f=\partial_{c}f\equiv \R$. In the Banach space of continuous functions defined on $[0,1]$, denoted by $C[0,1]$, with the uniform norm, we show that there exists a residual and prevalent set $D\subset C[0,1]$ such that $\partial_{a}f=\partial_{c}f\equiv \R$ on $[0,1]$ for every $f\in D$. In the space of automorphisms we prove that most functions $f$ satisfy $\partial_{a}f=\partial_{c}f\equiv [0,+\infty)$ on $[0,1]$. The subdifferentiability of the Weierstrass function and the Cantor function are completely analyzed. Similar results for Lipschitz functions are also given.