  In 1935 K. Kuratowski posed the problem whether a function f:X ! Y , (X is
completely metrizable and Y is metrizable), with the property that a preimage
of each open has the Baire property, is continuous apart from a meager set.
This paper is a selection of older results related to the question posed by
Kuratowski in 1935, coming from among others Solovay and Bukovsk?y, and quite
new ones concerning considerations in Ellentuck topology and some propertiesof
K-ideals, (i.e. ideals associated with Kuratowski partitions).
