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Title: Where are typical $C^{1}$ functions one-to-one? (English)
Author: Buczolich, Zoltán
Author: Máthé, András
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959 (print)
ISSN: 2464-7136 (online)
Volume: 131
Issue: 3
Year: 2006
Pages: 291-303
Summary lang: English
Category: math
Summary: Suppose $F\subset [0,1]$ is closed. Is it true that the typical (in the sense of Baire category) function in $C^{1}[0,1]$ is one-to-one on $F$? If ${\underline{\dim }}_{B}F<1/2$ we show that the answer to this question is yes, though we construct an $F$ with $\dim _{B}F=1/2$ for which the answer is no. If $C_{\alpha }$ is the middle-$\alpha $ Cantor set we prove that the answer is yes if and only if $\dim (C_{\alpha })\le 1/2.$ There are $F$’s with Hausdorff dimension one for which the answer is still yes. Some other related results are also presented. (English)
Keyword: typical function
Keyword: box dimension
Keyword: one-to-one function
MSC: 26A15
MSC: 28A78
MSC: 28A80
idZBL: Zbl 1112.26002
idMR: MR2248596
DOI: 10.21136/MB.2006.134143
Date available: 2009-09-24T22:26:37Z
Last updated: 2020-07-29
Stable URL:
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