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Search: MSC category 26A24 ( Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15] )

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1. CJM 2005 (vol 57 pp. 471)

Ciesielski, Krzysztof; Pawlikowski, Janusz
Small Coverings with Smooth Functions under the Covering Property Axiom
In the paper we formulate a Covering Property Axiom, \psmP, which holds in the iterated perfect set model, and show that it implies the following facts, of which (a) and (b) are the generalizations of results of J. Stepr\={a}ns. \begin{compactenum}[\rm(a)~~] \item There exists a family $\F$ of less than continuum many $\C^1$ functions from $\real$ to $\real$ such that $\real^2$ is covered by functions from $\F$, in the sense that for every $\la x,y\ra\in\real^2$ there exists an $f\in\F$ such that either $f(x)=y$ or $f(y)=x$. \item For every Borel function $f\colon\real\to\real$ there exists a family $\F$ of less than continuum many ``$\C^1$'' functions ({\em i.e.,} differentiable functions with continuous derivatives, where derivative can be infinite) whose graphs cover the graph of $f$. \item For every $n>0$ and a $D^n$ function $f\colon\real\to\real$ there exists a family $\F$ of less than continuum many $\C^n$ functions whose graphs cover the graph of $f$. \end{compactenum} We also provide the examples showing that in the above properties the smoothness conditions are the best possible. Parts (b), (c), and the examples are closely related to work of A. Olevski\v{\i}.

Keywords:continuous, smooth, covering
Categories:26A24, 03E35

2. CJM 1998 (vol 50 pp. 242)

Benoist, Joël
Intégration du sous-différentiel proximal: un contre exemple
Etant donn\'ee une partie $D$ d\'enombrable et dense de ${\R}$, nous construisons une infinit\'e de fonctions Lipschitziennes d\'efinies sur ${\R}$, s'annulant en z\'ero, dont le sous-diff\'erentiel proximal est \'egal \`a $]-1, 1[$ en tout point de $D$ et est vide en tout point du compl\'ementaire de $D$. Nous d\'eduisons que deux fonctions dont la diff\'erence n'est pas constante peuvent avoir les m\^emes sous-diff\'erentiels.

Categories:26A16, 26A24

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