Partitions into thin sets and forgotten theorems of Kunugi and Lusin-Novikov

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 a 1st category set. In 1935, Kuratowski asked whether the separability requirement could be lifted. 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 the fact that the first impressive attempt to solve the Kuratowski problem, due to Kinjiro Kunugi and based on a theorem of Lusin and Novikov, took place already in 1936. Lusin's remarkable 1934 Comptes Rendus note soon forgotten, remained unnoticed to this day. We analyze both papers and bring the results to full light.