# Universal chord theorem

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A chord (in red) of length 0.3 on a sinusoidal function. The universal chord theorem guarantees the existence of chords of length 1/n for functions satisfying certain conditions.

In mathematical analysis, the universal chord theorem states that if a function f is continuous on [a,b] and satisfies ${\displaystyle f(a)=f(b)}$, then for every natural number ${\displaystyle n}$, there exists some ${\displaystyle x\in [a,b]}$ such that ${\displaystyle f(x)=f(x+{\frac {1}{n}})}$.[1]

## History

The theorem was published by Paul Lévy in 1934 as a generalization of Rolle's Theorem.[2]

## Statement of the theorem

Let ${\displaystyle H(f)=\{h\in [0,1]:f(x)=f(x+h){\text{ for some }}x\}}$ denote the chord set of the function f. If f is a continuous function and ${\displaystyle h\in H(f)}$, then ${\displaystyle {\frac {h}{n}}\in H(f)}$ for all natural numbers n. [3]

## Case of n = 2

The case when n = 2 can be considered an application of the Borsuk–Ulam theorem to the real line. It says that if ${\displaystyle f(x)}$ is continuous on some interval ${\displaystyle I=[a,b]}$ with the condition that ${\displaystyle f(a)=f(b)}$, then there exists some ${\displaystyle x\in [a,b]}$ such that ${\displaystyle f(x)=f(x+1/2)}$.

In less generality, if ${\displaystyle f:[0,1]\rightarrow \mathbb {R} }$ is continuous and ${\displaystyle f(0)=f(1)}$, then there exists ${\displaystyle x\in \left[0,{\frac {1}{2}}\right]}$ that satisfies ${\displaystyle f(x)=f(x+1/2)}$.

## References

1. ^ Rosenbaum, J. T. (May, 1971) The American Mathematical Monthly, Vol. 78, No. 5, pp. 509–513
2. ^ Paul Levy, "Sur une Généralisation du Théorème de Rolle", C. R. Acad. Sci., Paris, 198 (1934) 424–425.
3. ^ Oxtoby, J.C. (May 1978). "Horizontal Chord Theorems". The American Mathematical Monthly. 79: 468–475. doi:10.2307/2317564.