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.


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