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