Hausdorff Dimension and Derivatives of Typical Nondecreasing Continuous Functions Alain Riviere To cite this version: Alain Riviere. Hausdorff Dimension and Derivatives of Typical Nondecreasing Continuous Functions. 2014. hal-01154558 HAL Id: hal-01154558 https://hal.archives-ouvertes.fr/hal-01154558 Preprint submitted on 22 May 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Hausdorff Dimension and Derivatives of Typical Nondecreasing Continuous Functions Alain RIVIERE 10 2014 Laboratoire Ami´enoisde Math´ematiquesFondamentales et Appliqu´ees, CNRS, UMR 7352 Facult´ede Math´ematiqueset Informatique d'Amiens 33 rue Saint-Leu, 80 039 Amiens Cedex 1, France.
[email protected] Abstract We endow the space Cr of nondecreasing functions on the unit interval [0; 1] with the uniform metric and consider its subspace Ccr of continuous nondecreasing functions. Then, we mainly prove: 1) In the sense of Baire categories, for most f 2 Ccr, and also for most f 2 Cr, we have: (a) the Stieljes measure mesf of f is carried by a set of Hausdorff dimension zero, (b) More precisely, f has zero left and right Diny lower derivatives everywhere outside a set of Hausdorff dimension zero, (c) For any 0 ≤ α ≤ 1, the set of all t 2 [0; 1] at which α is the left and right Diny upper derivative of f, is of Hausdorff dimension 1.