A Lipschitz function which is \(C^{\infty}\) on a.e. line need not be generically differentiable (Q2839275)
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scientific article; zbMATH DE number 6184248
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | A Lipschitz function which is \(C^{\infty}\) on a.e. line need not be generically differentiable |
scientific article; zbMATH DE number 6184248 |
Statements
4 July 2013
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Gâteaux differentiability
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Lipschitz function
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A Lipschitz function which is \(C^{\infty}\) on a.e. line need not be generically differentiable (English)
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The author constructs, for every Banach space \(X\) of dimension at least \(2\), a Lipschitz function \(f:X\to\mathbb R\) such that for each fixed nonzero vector \(v\in X\) restrictions of \(f\) to almost all lines parallel to \(v\) are \(C^\infty\) smooth, yet \(f\) is not generically Gâteaux differentiable, i.e., the set of points where \(f\) is Gâteaux differentiable is of the first category. In the case of finite-dimensional \(X\), the notion of almost all coincides with Lebesgue almost all; in the case that \(X\) is infinite dimensional, the exceptional set is contained in an Aronszajn null set and is also \(\Gamma\)-null.
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