On two consequences of CH established by Sierpiński, part 2

We study the relations between two consequences of the Continuum Hypothesis discovered by Wacław Sierpiński, concerning uniform continuity of continuous functions and uniform convergence of sequences of real-valued functions, defined on uncountable subsets of the real line. The subject is closely related  to the existence and properties of Lusin sets and  K-Lusin sets in the Baire space N^N, i.e., uncountable sets in N^N intersecting each compact set in N^N in an at most countable set.

The results come from two joint papers with Roman Pol, available at
http://arxiv.org/abs/2306.11712     and     http://arxiv.org/abs/2403.03110