Some measurability and continuity properties of arbitrary real functions
Given an arbitrary real function <em>f</em> , the set <em>D_f</em> of all points where <em>f</em> admits approximate limit is the maximal (with respect to the relation of inclusion except for a nullset) measurable subset of the real line having the properties that...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Università degli Studi di Catania
2002-05-01
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| Series: | Le Matematiche |
| Online Access: | http://www.dmi.unict.it/ojs/index.php/lematematiche/article/view/198 |
| Summary: | Given an arbitrary real function <em>f</em> , the set <em>D_f</em> of all points where <em>f</em> admits approximate limit is the maximal (with respect to the relation of inclusion except for a nullset) measurable subset of the real line having the properties that the restriction of <em>f </em>to <em>D_f </em>is measurable, and <em>f</em> is approximately continuous at almost every point of <em>D_f</em> . These results extend the well-known fact that a function is measurable if and only if it is approximately continuous almost everywhere. In addition, there exists a maximal <em>G_δ</em> -set <em>C_f</em> (which can be actually constructed from <em>f</em> ) such that it is possible to find a function<br /><em>g = f</em> almost everywhere, whose set of points of continuity is exactly <em>C_f</em> .<br /> |
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| ISSN: | 0373-3505 2037-5298 |