Restrictions of continuous functions

Jean-Pierre Kahane (1) , Yitzhak Katznelson (2)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 Stanford University

Given a continuous real-valued function on [0, 1], and a closed subset E ⊂ [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.

Mots clés: Minkowski and Hausdorff dimensions; Continuous functions; Lipschitz; Hölder; Bounded variation; Monotonicity; Elementary constructions and proofs.; Elementary constructions and proofs

Domaines: Analyse classique [math.CA]

Jean-Pierre Kahane, Yitzhak Katznelson. Restrictions of continuous functions. 2007. ⟨hal-00185028⟩