Connections between strong $I$-density (for plane sets) and one-dimensional $I$-density are studied. Other results deal with separately $I$-approximately continuous functions. A topology associated with separately $I$-approximately continuous functions, similar to O’Malley’s one is introduced and investigated. Among others it is shown that strong $I$-approximate continuity implies separate $I$-approximate continuity and that a separately $I$-approximately continuous function is Baire $2$ and need not be Baire $1$. Finally, some properties of functions with $I$-approximately continuous sections are studied.

I will show how non-standard analysis can help in describing numeration systems, such as that used by fixed-point arithmetics in computers. To achieve the non-standard extension of the total order, instead of the usual definition using ultrafilters, a lexicographical ordering will be used.