TY - JOUR

T1 - Continuous functions with impermeable graphs

AU - Buczolich, Zoltán

AU - Leobacher, Gunther

AU - Steinicke, Alexander

N1 - Publisher Copyright: © 2023 The Authors. Mathematische Nachrichten published by Wiley-VCH GmbH.

PY - 2023/10

Y1 - 2023/10

N2 - 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.

AB - 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.

KW - Hausdorff dimension of zeros

KW - uncountable zeros

KW - permeable sets

KW - permeable graph

KW - intrinsic metric

UR - http://www.scopus.com/inward/record.url?scp=85161561976&partnerID=8YFLogxK

U2 - 10.1002/mana.202200268

DO - 10.1002/mana.202200268

M3 - Article

VL - 296.2023

SP - 4778

EP - 4805

JO - Mathematische Nachrichten

JF - Mathematische Nachrichten

SN - 0025-584X

IS - 10

ER -