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1999/2000 Two Examples Concerning Extendable and Almost Continuous Functions
Krzysztof Ciesielski, Harvey Rosen
Real Anal. Exchange 25(2): 579-598 (1999/2000).

Abstract

The main purpose of this paper is to describe two examples. The first is that of an almost continuous, Baire class two, non-extendable function $f\colon[0,1]\to[0,1]$ with a $G_\delta$ graph. This answers a question of Gibson [15]. The second example is that of a connectivity function $F\colon\mathbb{R}^2\to\mathbb{R}$ with dense graph such that $F^{-1}(0)$ is contained in a countable union of straight lines. This easily implies the existence of an extendable function $f\colon\mathbb{R}\to\mathbb{R}$ with dense graph such that $f^{-1}(0)$ is countable. We also give a sufficient condition for a Darboux function $f\colon[0,1]\to[0,1]$ with a $G_\delta$ graph whose closure is bilaterally dense in itself to be quasi-continuous and extendable.

Citation

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Krzysztof Ciesielski. Harvey Rosen. "Two Examples Concerning Extendable and Almost Continuous Functions." Real Anal. Exchange 25 (2) 579 - 598, 1999/2000.

Information

Published: 1999/2000
First available in Project Euclid: 3 January 2009

zbMATH: 1010.26002
MathSciNet: MR1778512

Subjects:
Primary: 26A15
Secondary: 54A25

Keywords: extendable functions , peripherally continuous functions

Rights: Copyright © 1999 Michigan State University Press

Vol.25 • No. 2 • 1999/2000
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