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Suppose F ⊂ [0, 1] is closed. Is it true that the typical (in the sense of Baire category) function in C1[0, 1] is one-to-one on F? If dimBF < 1/2 we show that the answer to this question is yes, though we construct an F with dimBF = 1/2 for which the answer is no. If Cα is the middle-α Cantor set we prove that the answer is yes if and only if dim(Cα) ≤ 1/2. There are F ’s with Hausdorff dimension one for which the answer is still yes. Some other related results are also presented. Research supported by the Hungarian National Foundation for Scientific Research T049727. Supported by the Hungarian Scientific Research Fund grant no. T049786. 2000 Mathematics Subject Classification: Primary 26A15; Secondary 28A78, 28A80.

Where are typical $C^{1}$ functions one-to-one?