A note on uniform continuity of monotone functions

Abstract

We prove that it is consistent with ZFC that for every non-decreasing function $f:[0,1]\to [0,1]$, each subset of $[0,1]$ of cardinality $\mathfrak{c}$ contains a set of cardinality $\mathfrak{c}$ on which $f$ is uniformly continuous. We show that this statement follows from the assumptions that ${\mathfrak{d}}^{\ast }<\mathfrak{c}$ and $\mathfrak{c}$ is regular, where ${\mathfrak{d}}^{\ast }\le \mathfrak{d}$ is the smallest cardinality $\kappa$ such that any two disjoint countable dense sets in the Cantor set can be separated by sets each of which is an intersection of at most $\kappa$-many open sets in the Cantor set. We establish also that ${\mathfrak{d}}^{\ast }=\min \{\mathfrak{u},\mathfrak{d}\}=\min \{\mathfrak{r},\mathfrak{d}\}$, thus giving an alternative proof of the latter equality established by J. Aubrey in 2004.