Open Access
2004-2005 Subdifferentiability of real functions
Xianfu Wang
Real Anal. Exchange 30(1): 137-172 (2004-2005).


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.


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Xianfu Wang. "Subdifferentiability of real functions." Real Anal. Exchange 30 (1) 137 - 172, 2004-2005.


Published: 2004-2005
First available in Project Euclid: 27 July 2005

zbMATH: 1061.49016
MathSciNet: MR2127522

Primary: 26A24
Secondary: 26A30 , 26A48

Keywords: absolutely continuous function , automorphism , Baire category , Continuous function , Dini derivative , Lebesgue measure zero , Lipschitz function , monotone function , nowhere monotone function of a second specie , subdifferential

Rights: Copyright © 2004 Michigan State University Press

Vol.30 • No. 1 • 2004-2005
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