Abstract
We investigate how large a set can be on which a continuous nowhere monotone function is one-to-one. We consider the σ-ideal of microscopic sets, which is situated between the countable sets and the sets of Hausdorff dimension zero and prove that the typical function in C[0, 1] (in the sense of Baire) is nowhere monotone and one-to-one except on some microscopic set. We also give an example of a continuous nowhere monotone function of bounded variation on [0, 1], which is one-to-one except on some microscopic set, so it is not a typical function.
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Karasińska, A., Wagner-Bojakowska, E. Nowhere monotone functions and microscopic sets. Acta Math Hung 120, 235–248 (2008). https://doi.org/10.1007/s10474-008-7093-y
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DOI: https://doi.org/10.1007/s10474-008-7093-y