Real Analysis Exchange

On the chord set of continuous functions

Marianna Csörnyei

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Abstract

It is well-known that for a given continuous function \(f\), \(f(0)=f(1)\) and for any natural number \(n\) there exist \(x_n\), \(y_n=x_n + 1/n\) such that \(f(x_n)=f(y_n)\). It is also known that if the graph of \(f\) (or more generally a planar curve connecting the point 0 and 1) does not have a horizontal chord of length \(a\) and \(b\) respectively then there is no horizontal chord of length \(a+b\) either (see \cite{YY}). It is almost immediate that the lengths of possible horizontal chords of \(f\) form a closed set \(F\) of the unit interval [0,1], and according to the remark above its complement \(G=[0,1]\setminus F\) is an additive set: \(a\in G\), \(b\in G\), \(a+b\le 1\) imply \(a+b \in G\). C. Ryll-Nardzewski, Z. Romanowicz and M. Morayne raised the problem whether this additive property is not just necessary but also sufficient for a set to be the complement of the chord-set of some continuous function. In this paper we answer their question affirmatively.

Article information

Source
Real Anal. Exchange, Volume 22, Number 2 (1996), 853-855.

Dates
First available in Project Euclid: 22 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.rae/1337713166

Mathematical Reviews number (MathSciNet)
MR1460997

Zentralblatt MATH identifier
0940.26002

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}

Keywords
continuous functions chord length

Citation

Csörnyei, Marianna. On the chord set of continuous functions. Real Anal. Exchange 22 (1996), no. 2, 853--855. https://projecteuclid.org/euclid.rae/1337713166


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References

  • A. M. Yaglom and I. M. Yaglom, Non-elementary problems in elementary presentation, GITTL, Moscow, 1954, Problem 118, p. 60 (in Russian).