Concerning a characterization of continuity

Abstract

Two problems related to the characterization of continuity are discussed. In the first problem “\(f\) is almost continuous in the sense of Stallings” will be replaced with a weaker condition “\(f\) is a Darboux function” and it will be shown that the characterization of continuity remains true. Also it follows that for the classes of functions considered, “\(f\) is a Darboux function” is the weakest possible condition for which the characterization remains true. In the second problem “\(f\) is almost continuous in the sense of Stallings” will be replaced with a stronger condition “\(f\) is an extendable function”. Then it will be shown that this condition and conditions (2) and (3) are not redundant.