The one-to-one restrictions of functions

Rok, strany: 2008, 161 - 169

A continuous and nowhere monotone function for which every set having the Baire property, on which it is one-to-one, is of first category, is constructed here. Further, it is shown that every continuous and nowhere monotone function has the above property. We also show that an analogous result fails to hold if one does not assume that a set on which the function is one-to-one is a Baire set. \par In the second part of the paper we investigate the Lebesgue measure of a set on which a continuous nowhere monotone function is one-to-one. Here the situation turns out to be more varied. For each $η\in[0,1)$ we construct a continuous function non-monotone on any interval with the following property: there exists a set of measure $η$ on which this function is one-to-one, and every set on which the function is one-to-one has measure smaller or equal to $η$.

ISO 690:

Karasińska, A. 2008. The one-to-one restrictions of functions. In Tatra Mountains Mathematical Publications, vol. 40, no.2, pp. 161-169. 1210-3195.

APA:

Karasińska, A. (2008). The one-to-one restrictions of functions. Tatra Mountains Mathematical Publications, 40(2), 161-169. 1210-3195.