Continuous functions with impermeable graphs

Zoltán Buczolich, Gunther Leobacher, Alexander Steinicke

Publikation: Beitrag in FachzeitschriftArtikelForschungBegutachtung

Abstract

We construct a Hölder continuous function on the unit interval which coincides in uncountably (in fact continuum) many points with every function of total variation smaller than 1 passing through the origin. We conclude that this function has impermeable graph—one of the key concepts introduced in this paper—and we present further examples of functions both with permeable and impermeable graphs. Moreover, we show that typical (in the sense of Baire category) continuous functions have permeable graphs. The first example function is subsequently used to construct an example of a continuous function on the plane which is intrinsically Lipschitz continuous on the complement of the graph of a Hölder continuous function with impermeable graph, but which is not Lipschitz continuous on the plane. As another main result, we construct a continuous function on the unit interval which coincides in a set of Hausdorff dimension 1 with every function of total variation smaller than 1 which passes through the origin.

OriginalspracheEnglisch
Seiten (von - bis)4778-4805
Seitenumfang28
FachzeitschriftMathematische Nachrichten
Jahrgang296.2023
Ausgabenummer10
Frühes Online-Datum7 Juni 2023
DOIs
PublikationsstatusVeröffentlicht - Okt. 2023

Bibliographische Notiz

Funding Information:
Z. Buczolich: The project leading to this application has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 741420). This author was also supported by the Hungarian National Research, Development and Innovation Office–NKFIH, Grant 124749 and at the time of completion of this paper was holding a visiting researcher position at the Rényi Institute. G. Leobacher is supported by the Austrian Science Fund (FWF): Project F5508‐N26, which is part of the Special Research Program “Quasi‐Monte Carlo Methods: Theory and Applications.”

Funding Information:
Gunther Leobacher and Alexander Steinicke are grateful to Dave L. Renfro for connecting them with Zoltán Buczolich. Z. Buczolich: The project leading to this application has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 741420). This author was also supported by the Hungarian National Research, Development and Innovation Office–NKFIH, Grant 124749 and at the time of completion of this paper was holding a visiting researcher position at the Rényi Institute. G. Leobacher is supported by the Austrian Science Fund (FWF): Project F5508-N26, which is part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications.”

Publisher Copyright:
© 2023 The Authors. Mathematische Nachrichten published by Wiley-VCH GmbH.

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