Real Analysis Exchange

Subdifferentiability of real functions

Xianfu Wang

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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.

Article information

Real Anal. Exchange, Volume 30, Number 1 (2004), 137-172.

First available in Project Euclid: 27 July 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15]
Secondary: 26A30: Singular functions, Cantor functions, functions with other special properties 26A48: Monotonic functions, generalizations

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


Wang, Xianfu. Subdifferentiability of real functions. Real Anal. Exchange 30 (2004), no. 1, 137--172.

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