Étude sur la détermination d'une fonction discontinue par sa dérivée unilatérale

Viola, Tullio

doi:10.24033/asens.826

Cited by 2 publications

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“…Hobson [7] in his account of the period used the notions but failed to employ any terminology (even avoiding the French term which, by then, was well known). Viola [10] (a student of Denjoy) is perhaps the first to study extensively the one sided…”

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confidence: 99%

“…The remainder of this article is devoted to further applications of Proposition 1 showing how it provides a basic tool for establishing that many exceptional sets are scattered or semi-scattered. Our first application is to a characterization from Viola [10] of semiscattered sets. Our second example is a simple semiscattered property of real functions.…”

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confidence: 99%

“…This may be found in various forms elsewhere. For example, if we let g denote the oscillation function of f , then it includes Viola's [10] observation that a function f with a left hand derivative must be continuous off a right scattered set; Charzyński [2] employs a similar argument in a different context.…”

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Scattered Sets and Gauges

Freiling

Thomson

1995

Real Analysis Exchange

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An elementary and natural method for demonstrating that certain exceptional sets are scattered is presented.In this note we wish to present a simple technique that can be used to establish that certain exceptional sets are scattered.Recall that a set of real numbers is scattered if every nonempty subset has an isolated point. One sided versions have been considered in the past: a set is right [left] scattered if every nonempty subset has a point isolated on the right [left]; any such set is called semi-scattered. A set is splattered if every nonempty subset has a point isolated on one side at least. A splattered set may be expressed as the union of a right scattered set and a left scattered set. Scattered sets may, similarly, be viewed as the intersection of a right scattered set and a left scattered set.The first explicit use of such ideas is in Cantor [1] where he uses the term separierte Mengen for a set that contains no subset dense-in-itself. In the first decades of this century, G. C. Young and W. H. Young made considerable use of scattered sets including left and right versions but employed no terminology. Denjoy introduced the term clairsemé into French language accounts and Hausdorff employed zerstreute Mengen in his writing. Hobson [7] in his account of the period used the notions but failed to employ any terminology (even avoiding the French term which, by then, was well known). Viola [10] (a student of Denjoy) is perhaps the first to study extensively the one sided

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confidence: 99%