Title:

Where are typical $C^{1}$ functions onetoone? (English) 
Author:

Buczolich, Zoltán 
Author:

Máthé, András 
Language:

English 
Journal:

Mathematica Bohemica 
ISSN:

08627959 (print) 
ISSN:

24647136 (online) 
Volume:

131 
Issue:

3 
Year:

2006 
Pages:

291303 
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 onetoone 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:

onetoone function 
MSC:

26A15 
MSC:

28A78 
MSC:

28A80 
idZBL:

Zbl 1112.26002 
idMR:

MR2248596 
DOI:

10.21136/MB.2006.134143 
. 
Date available:

20090924T22:26:37Z 
Last updated:

20200729 
Stable URL:

http://hdl.handle.net/10338.dmlcz/134143 
. 
Reference:

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Reference:

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. 