Some remarks on $\rho$-upper continuous functions

Abstract

The notion of a $\rho$-upper continuous function is a  generalization of the notion of an approximately continuous function. It was introduced by S. Kowalczyk and K. Nowakowska. In \cite {kn} the authors proved that each $\rho$-upper continuous function is measurable and has Denjoy property. In this note we prove that there exists a measurable function having Denjoy property which is not $\rho$-upper continuous function for any $\rho\in[0,1)$ and there exists a function which is $\rho$-upper continuous for each $\rho \in [0,1)$ and is not approximately continuous. In \cite {kn} the authors also proved  that for each $\rho\in(0,\frac{1}{2})$ there exists a $\rho$-upper continuous function which is not in the first class of Baire. Here we show that there exists a function which is $\rho$-upper continuous for each $\rho\in[0,1)$ but is not Baire 1 function.