  A function F:R^2->R is sup-measurable if F_f:R->R given by F_f(x)=F(x,f(x)),
x in R, is measurable for each measurable function f:R->R. It is known that
under different set theoretical assumptions, including CH, there are
sup-measurable non-measurable functions, as well as their category analog. In
this paper we will show that the existence of category analog of sup-measurable
non-measurable functions is independent of ZFC. A similar result for the
original measurable case is a subject of a work in prepartion by Roslanowski
and Shelah.
