  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.
