The sets where a function has infinite one-sided derivatives (Q810649)
From MaRDI portal
 | This is the item page for this Wikibase entity, intended for internal use and editing purposes. Please use this page instead for the normal view: The sets where a function has infinite one-sided derivatives |
scientific article; zbMATH DE number 4214277
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | The sets where a function has infinite one-sided derivatives |
scientific article; zbMATH DE number 4214277 |
Statements
The sets where a function has infinite one-sided derivatives (English)
0 references
1991
0 references
The main result is the following: Let \(E_ 1\), \(E_ 2\) be disjoint subsets of \({\mathbb{R}}\). There exists a function \(f: {\mathbb{R}}\to {\mathbb{R}}\) such that \(E_ 1=\{x:\;f'_ -(x)=\infty \}\) and \(E_ 2=\{x:\;f'_ - (x)=-\infty \}\) if and only if \(m(E_ 1)=m(E_ 2)=0\) \((f'_ -(x)\) the left-hand derivative at x, m the Lebesgue measure).
0 references
right-hand derivative
0 references
set of measure zero
0 references
Borel set
0 references
Denjoy-Young-Saks theorem
0 references
left-hand derivative
0 references
