## Real Analysis Exchange

### Darboux Quasicontinuous Functions

Harvey Rosen

#### 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.

#### Article information

Source
Real Anal. Exchange, Volume 23, Number 2 (1999), 631-640.

Dates
First available in Project Euclid: 14 May 2012