Nowhere monotone functions and microscopic sets

We investigate how large a set can be on which a continuous nowhere monotone function is one-to-one. We consider the sigma-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.