Darboux Quasicontinuous Functions

Abstract

Let \(C(f)\) denote the set of points at which a function \(f:I\to I\) is continuous, where \(I=[0,1]\). We show that if a Darboux quasicontinuous function \(f\) has a graph whose closure is bilaterally dense in itself, then \(f\) is extendable to a connectivity function \(F: I^2\to I\) and the set \(I\setminus C(f)\) of points of discontinuity of \(f\) is \(f\)-negligible. We also show that although the family of Baire class 1 quasicontinuous functions can be characterized by preimages of sets, the family of Darboux quasicontinuous functions cannot. An example is found of an extendable function \(f: I\to \mathbb{R}\) which is not of Cesaro type and not quasicontinuous.