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Register a new userMeasurability of functions with approximately continuous vertical sections and measurable horizontal sections
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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.
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The linear continuity of a function defined on a vector space means that its restriction on every affine line is continuous. For functions defined on $mathbb R^m$ this notion is near to the separate continuity for which it is required only the continuity on the straight lines which are parallel to coordinate axes. The classical Lebesgue theorem states that every separately continuous function $f:mathbb R^mtomathbb R$ is of the $(m-1)$-th Baire class. In this paper we prove that every linearly continuous function $f:mathbb R^mtomathbb R$ is of the first Baire class. Moreover, we obtain the following result. If $X$ is a Baire cosmic topological vector space, $Y$ is a Tychonoff topological space and $f:Xto Y$ is a Borel-measurable (even BP-measurable) linearly continuous function, then $f$ is $F_sigma$-measurable. Using this theorem we characterize the discontinuity point set of an arbitrary linearly continuous function on $mathbb R^m$. In the final part of the article we prove that any $F_sigma$-measurable function $f:partial Uto mathbb R$ defined on the boundary of a strictly convex open set $Usubsetmathbb R^m$ can be extended to a linearly continuous function $bar f:Xto mathbb R$. This fact shows that in the ``descriptive sense the linear continuity is not better than the $F_sigma$-measurability.
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A function f from reals to reals (f:R->R) is almost continuous (in the sense of Stallings) iff every open set in the plane which contains the graph of f contains the graph of a continuous function. Natkaniec showed that for any family F of continuum many real functions there exists g:R->R such that f+g is almost continuous for every f in F. Let AA be the smallest cardinality of a family F of real functions for which there is no g:R->R with the property that f+g is almost continuous for every f in F. Thus Natkaniec showed that AA is strictly greater than the continuum. He asked if anything more could be said. We show that the cofinality of AA is greater than the continuum, c. Moreover, we show that it is pretty much all that can be said about AA in ZFC, by showing that AA can be equal to any regular cardinal between c^+ and 2^c (with 2^c arbitrarily large). We also show that AA = AD where AD is defined similarly to AA but for the class of Darboux functions. This solves another problem of Maliszewski and Natkaniec.
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