Continuous functions with impermeable graphs
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In: Mathematische Nachrichten, Vol. 296.2023, No. 10, 10.2023, p. 4778-4805.
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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 -