## 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

### Keywords:

typical function; box dimension; one-to-one function
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