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Where are typical \(C^{1}\) functions one-to-one? (English) Zbl 1112.26002

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 \(\underline {\text{dim}} _{B}F<1/2\), we show that the answer to this question is yes, though we construct an \(F\) with \(\text{dim} _{B}F=1/2\) 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 \(\text{dim} (C_{\alpha })\leq 1/2.\) There are \(F\)’s with Hausdorff dimension one for which the answer is still yes. Some other related results are also presented.

MSC:

26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
28A78 Hausdorff and packing measures
28A80 Fractals
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