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 $\mathfrak d^* < \mathfrak c$ and $\mathfrak c$ is regular, where $\mathfrak d^*\leq \mathfrak d$ 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 $\mathfrak d^*=\min\{\mathfrak u, \mathfrak d\}=\min\{\mathfrak r, \mathfrak d\}$, thus giving an alternative proof of the latter equality established by J. Aubrey in 2004.