Real Analysis Exchange

Darboux like functions

Richard G. Gibson and Tomasz Natkaniec

Full-text: Open access

Abstract

A function \(f:\mathbb{R}\to\mathbb{R}\) is said to have the intermediate value property provided that if \(p\) and \(q\) are real numbers such that \(p\neq q\) and \(f(p)\lt f(q)\), then for every \(y\in (f(p),f(q))\) there exists a number \(x\) between \(p\) and \(q\) with \(f(x)=y\). In 1875, G. Darboux showed that there exist functions with the intermediate value property that are not continuous \cite{22}. Because of his work with functions having the intermediate value property, these functions are called Darboux functions.

Article information

Source
Real Anal. Exchange, Volume 22, Number 2 (1996), 492-533.

Dates
First available in Project Euclid: 22 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.rae/1337713140

Mathematical Reviews number (MathSciNet)
MR1460971

Zentralblatt MATH identifier
0942.26004

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 functions extendable functions almost continuous functions connectivity functions functions with perfect road peripherally continuous functions DIVP-functions CIVP-functions SCIVP-functions WCIVP-functions Sierpiński-Zygmund functions property (\(B\))

Citation

Gibson, Richard G.; Natkaniec, Tomasz. Darboux like functions. Real Anal. Exchange 22 (1996), no. 2, 492--533. https://projecteuclid.org/euclid.rae/1337713140


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