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Differentiability a.e. and approximate differentiability a.e
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Titel: |
Differentiability a.e. and approximate differentiability a.e |
In: | Proceedings of the American Mathematical Society, 66, 1977, 2, S. 294-298 |
veröffentlicht: |
American Mathematical Society (AMS)
|
Umfang: | 294-298 |
ISSN: |
0002-9939 1088-6826 |
DOI: | 10.1090/s0002-9939-1977-0453938-0 |
Zusammenfassung: | <p>Let <italic>F</italic> be a finite real valued function defined on [0, 1]. We prove that <italic>F</italic> can be transformed into a function which is differentiable a.e. by a homeomorphic change of variables if and only if <italic>F</italic> is continuous on a dense set. We also show that <italic>F</italic> can be transformed into a function which is approximately differentiable a.e. if and only if each interval <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I subset-of left-bracket 0 comma 1 right-bracket"> <mml:semantics> <mml:mrow> <mml:mi>I</mml:mi> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">I \subset [0,1]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> contains a nonempty perfect set <italic>P</italic> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F vertical-bar upper P"> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">F|P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is continuous.</p> |
Format: | E-Article |
Quelle: | American Mathematical Society (AMS) (CrossRef) |
Sprache: | Englisch |