Some observations on the symmetrical quasicontinuity of Piotrowski and Vallin (Q2497020)

The author considers functions \(f:\mathbb R^2\rightarrow\mathbb R\) with respect to the Euclidean topology and with respect to the density topology and investigates the Lebesgue measurability of these functions assuming some quasicontinuity-type conditions. It is pointed out that quasicontinuous functions are cliquish and evidently symmetrically quasicontinuous functions are quasicontinuous and symmetrically cliquish. Then the author proves that, with the Euclidean topology, there are symmetrically quasicontinuous functions \(f\) which are not measurable. On the other hand, it is also proved that, with the density topology, all symmetrically cliquish functions \(f\) are measurable.