Partitions into thin sets and forgotten theorems of Kunugi and Luzin–Novikov
Volume 155 / 2019
Colloquium Mathematicum 155 (2019), 267-285
MSC: 28A05, 54C10, 54E52.
DOI: 10.4064/cm7287-4-2018
Published online: 10 December 2018
Abstract
Let $f$ be a function from a metric space $Y$ to a separable metric space $X$. If $f$ has the Baire property, then it is continuous apart from a 1st category set. In 1935, Kuratowski asked whether the separability requirement could be removed. A full scale attack on the problem took place in the late seventies and early eighties. What was not known then, and what remains virtually unknown today, is that a first impressive attempt to solve the Kuratowski problem, due to Kinjiro Kunugi and based on a theorem of Luzin and Novikov, had already taken place in 1936. Luzin’s remarkable 1934 Comptes Rendus note, soon forgotten, has remained unnoticed to this day. We analyze both papers and bring the results to full light.
