Continuous functions with impermeable graphs. (English) Zbl 07750741
Summary: 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.
© 2023 Wiley-VCH GmbH.
© 2023 Wiley-VCH GmbH.
MSC:
| 26A16 | Lipschitz (Hölder) classes |
| 28A78 | Hausdorff and packing measures |
| 26A21 | Classification of real functions; Baire classification of sets and functions |
| 26B30 | Absolutely continuous real functions of several variables, functions of bounded variation |
| 26B35 | Special properties of functions of several variables, Hölder conditions, etc. |
| 54C05 | Continuous maps |
| 54E40 | Special maps on metric spaces |
Keywords:
Hausdorff dimension of zeros; intrinsic metric; permeable graph; permeable sets; uncountable zerosReferences:
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