Mathematics – Classical Analysis and ODEs
Scientific paper
2007-11-05
Mathematics
Classical Analysis and ODEs
Article soumis \`a Israel Journal of Mathematics
Scientific paper
Given a continuous real-valued function on [0, 1], and a closed subset E \subset [0, 1] we denote by f E the restriction of f to E, that is, the function defined only on E that takes the same values as f at every point of E >. The restriction f E will typically be "better behaved" than f . It may have bounded variation when f doesn't, it may have a better modulus of continuity than f, it may be monotone when f is not, etc. All this clearly depends on f and on E, and the questions that we discuss here are about the existence, for every f, or every f in some class, of "substantial" sets E such that f E has bounded total variation, is monotone, or satisfies a given modulus of continuity. The notion of "substantial" that we use is that of either Hausdorff or Minkowski dimensions.
Kahane Jean-Pierre
Katznelson Yitzhak
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