Real Analysis Exchange

Continuity of Darboux Functions with Nice Finite Iterations

Kandasamy Muthuvel

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Abstract

A function that maps intervals into intervals is called a Darboux function. We prove that if $g$ is a continuous function that is non-constant on every non-empty open interval, and $f$ is a Darboux function such that, for every real number $x,$ $f^{n_{x}}(x)=g(x)$ for some positive integer $n_{x}$, and the set of all such $n_{x}$ is bounded, then $f$ is continuous. In the above statement, the hypothesis ``the set of all such $ n_{x} $ is bounded'' cannot be dropped. We also show that if $g$ is a continuous function that takes a constant value $k$ on some non-empty open interval $I$ and $k\in I$, then there exists a discontinuous Darboux function $f\:mathbb{R}\rightarrow \mathbb{R}$ with the property that, for every real number $x,$ $f^{n_{x}}(x)=g(x)$ for some positive integer $n_{x}\leq 2$. In the previous statement, if $k\notin I$, then no conclusion can be drawn about the function $f$.

Article information

Source
Real Anal. Exchange, Volume 32, Number 2 (2006), 587-596.

Dates
First available in Project Euclid: 3 January 2008

Permanent link to this document
https://projecteuclid.org/euclid.rae/1199377496

Mathematical Reviews number (MathSciNet)
MR2369868

Zentralblatt MATH identifier
1130.26001

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 n-to-1 functions continuous functions

Citation

Muthuvel, Kandasamy. Continuity of Darboux Functions with Nice Finite Iterations. Real Anal. Exchange 32 (2006), no. 2, 587--596. https://projecteuclid.org/euclid.rae/1199377496


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