## Real Analysis Exchange

### On the chord set of continuous functions

Marianna Csörnyei

#### 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

https://projecteuclid.org/euclid.rae/1337713166

Mathematical Reviews number (MathSciNet)
MR1460997

Zentralblatt MATH identifier
0940.26002

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

#### References

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