Real Analysis Exchange

Two Examples Concerning Extendable and Almost Continuous Functions

Krzysztof Ciesielski and Harvey Rosen

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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.

Article information

Source
Real Anal. Exchange, Volume 25, Number 2 (1999), 579-598.

Dates
First available in Project Euclid: 3 January 2009

Permanent link to this document
https://projecteuclid.org/euclid.rae/1230995394

Mathematical Reviews number (MathSciNet)
MR1778512

Zentralblatt MATH identifier
1010.26002

Subjects
Primary: 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) {For properties determined by Fourier coefficients, see 42A16; for those determined by approximation properties, see 41A25, 41A27}
Secondary: 54A25: Cardinality properties (cardinal functions and inequalities, discrete subsets) [See also 03Exx] {For ultrafilters, see 54D80}

Keywords
extendable functions peripherally continuous functions

Citation

Ciesielski, Krzysztof; Rosen, Harvey. Two Examples Concerning Extendable and Almost Continuous Functions. Real Anal. Exchange 25 (1999), no. 2, 579--598. https://projecteuclid.org/euclid.rae/1230995394


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