## Real Analysis Exchange

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

### On the chord set of continuous functions

#### 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

**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