{
  "id": "0711.0679",
  "version": "v1",
  "published": "2007-11-05T16:20:39.000Z",
  "updated": "2007-11-05T16:20:39.000Z",
  "title": "Restrictions of continuous functions",
  "authors": [
    "Jean-Pierre Kahane",
    "Yitzhak Katznelson"
  ],
  "comment": "Article soumis \\`a Israel Journal of Mathematics",
  "categories": [
    "math.CA"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-11-05T16:20:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26A15",
      "26A16",
      "26A45",
      "26A48"
    ],
    "keywords": [
      "continuous functions",
      "restriction",
      "bounded total variation",
      "substantial",
      "minkowski dimensions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0711.0679K"
    }
  }
}