  A function f:R -> R is approximately continuous iff it is continuous in the
density topology, i.e., for any ordinary open set U the set E=f^{-1}(U) is
measurable and has Lebesgue density one at each of its points. Denjoy proved
that approximately continuous functions are Baire 1., i.e., pointwise For any
f:R^2 -> R define f_x(y) = f^y(x) = f(x,y). A function f:R^2 -> R is separately
continuous if f_x and f^y are continuous for every x,y in R. Lebesgue in his
first paper proved that any separately continuous function is Baire 1.
Sierpinski showed that there exists a nonmeasurable f:R^2 -> R which is
separately Baire 1.
  In this paper we prove:
  Thm 1. Let f:R^2 -> R be such that f_x is approximately continuous and f^y is
Baire 1 for every x,y in R. Then f is Baire 2.
  Thm 2. Suppose there exists a real-valued measurable cardinal. Then for any
function f:R^2 -> R and countable ordinal i, if f_x is approximately continuous
and f^y is Baire i for every x,y in R, then f is Baire i+1 as a function of two
variables.
  Thm 3. (i) Suppose that R can be covered by omega_1 closed null sets. Then
there exists a nonmeasurable function f:R^2 -> R such that f_x is approximately
continuous and f^y is Baire 2 for every x,y in R. (ii) Suppose that R can be
covered by omega_1 null sets. Then there exists a nonmeasurable function f:R^2
-> R such that f_x is approximately continuous and f^y is Baire 3 for every x,y
in R.
  Thm 4. In the random real model for any function f:R^2 -> R if f_x is
approximately continuous and f^y is measurable for every x,y in R, then f is
measurable as a function of two variables.
