Real Analysis Exchange

Darboux Like Functions that are Characterizable by Images, Preimages and Associated Sets

Krzysztof Ciesielski

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Abstract

For \(\mathcal{A},\mathcal{B}\subset\mathcal{P}(\mathbb{R})\) let \(\mathcal{C}_{\mathcal{A},\mathcal{B}}=\{ f\in\mathbb{R}^\mathbb{R}\colon(\forall A\in\mathcal{A})\,(f(A)\in\mathcal{B})\}\) and \(\mathcal{C}_{\mathcal{A},\mathcal{B}}^{-1}=\{ f\in\mathbb{R}^\mathbb{R}\colon(\forall B\in\mathcal{B})\,(f^{-1}(B)\in\mathcal{A})\}\). A family \(\mathcal{F}\) of real functions is characterizable by images (preimages) of sets if \(\mathcal{F}=\mathcal{C}_{\mathcal{A},\mathcal{B}}\) (\(\mathcal{F}=\mathcal{C}_{\mathcal{A},\mathcal{B}}^{-1}\), respectively) for some \(\mathcal{A},\mathcal{B}\subset\mathcal{P}(\mathbb{R})\). We study which of the classes of Darboux like functions can be characterized in this way. Moreover, we prove that the class of all Sierpiński-Zygmund functions can be characterized by neither images nor preimages of sets.

Article information

Source
Real Anal. Exchange, Volume 23, Number 2 (1999), 441-458.

Dates
First available in Project Euclid: 14 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.rae/1337001357

Mathematical Reviews number (MathSciNet)
MR1639944

Zentralblatt MATH identifier
0943.26007

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} {Sierpi{ń}ski-Zygmund functions} {associated sets}

Citation

Ciesielski, Krzysztof. Darboux Like Functions that are Characterizable by Images, Preimages and Associated Sets. Real Anal. Exchange 23 (1999), no. 2, 441--458. https://projecteuclid.org/euclid.rae/1337001357


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