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MATHEMATICA BOHEMICA, Vol. 131, No. 3, pp. 291-303 (2006)
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Where are typical $C^{1}$ functions one-to-one?

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Zoltan Buczolich, Andras Mathé

* Zoltan Buczolich*, Department of Analysis, Eötvös Lorand University, Pazmany Péter Sétany 1/c, 1117 Budapest, Hungary, e-mail: ` buczo@cs.elte.hu`, {\tt www.cs.elte.hu/ {}buczo}; * Andras Mathé*, Department of Analysis, Eötvös Lorand University, Pazmany Péter Sétany 1/c, 1117 Budapest, Hungary, e-mail: ` amathe@cs.elte.hu`, ` amathe.web.elte.hu`

**Abstract:** Suppose $F\sse [0,1]$ 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 $\ldim _{B}F<1/2$ 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 $C_{\aaa }$ is the middle-$\aaa $ Cantor set we prove that the answer is yes if and only if $\dim (C_{\aaa })\leq 1/2.$ There are $F$'s with Hausdorff dimension one for which the answer is still yes. Some other related results are also presented.

**Keywords:** typical function, box dimension, one-to-one function

**Classification (MSC2000):** 26A15, 28A78, 28A80

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