A note on uniform continuity of monotone functions
Roman Pol, Piotr Zakrzewski, Lyubomyr Zdomskyy
TL;DR
The paper investigates the independence of the uniform-continuity statement (*) for monotone functions on large subsets of [0,1] and introduces the cardinal d^* to connect this question with classical continuum invariants. It shows that (*) is consistent under d^*<c together with c regular and proves the exact equalities d^* = min{u,d} = min{r,d}, using semifilter and ultrafilter arguments. It also discusses models where d^*<d (e.g., Miller and Blass–Shelah) and proposes a natural generalization d^*(X,D0,D1) for compact spaces, highlighting the topological meaning of these invariants. Overall, the work clarifies how set-theoretic assumptions shape uniform continuity properties on large sets and provides tools linking topology with cardinal characteristics of the continuum.
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 $κ$ 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 $κ$-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.