Borsik's properties of topological spaces and their applications

Let X be an uncountable Polish space. Lubica Hola showed recently that there are 2^continuum many quasi-continuous real valued functions defined on the uncountable Polish space that are not Borel measurable. Inspired by Hola’s result, we are extending it in two directions. First, we prove that the same conclusion holds when X is an uncountable Polish space and Y is any Hausdorff space with |Y|>1 then the family of all non-Borel measurable quasi-continuous functions has cardinality at least 2^continuum. Secondly, we show that the family of quasi-continuous non Borel functions may contain big algebraic structures.