Real Analysis Exchange

Darboux Quasicontinuous Functions

Harvey Rosen

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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

Permanent link to this document
https://projecteuclid.org/euclid.rae/1337001370

Mathematical Reviews number (MathSciNet)
MR1639996

Zentralblatt MATH identifier
0943.26010

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: 54C30: Real-valued functions [See also 26-XX]

Keywords
{Darboux function} {quasicontinuous function} {extendable connectivity function} {Baire 1 function} {characterizable by preimages} {Cesaro type function}

Citation

Rosen, Harvey. Darboux Quasicontinuous Functions. Real Anal. Exchange 23 (1999), no. 2, 631--640. https://projecteuclid.org/euclid.rae/1337001370


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