Universal chord theorem

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 $f(a)=f(b)$, then for every natural number $n$, there exists some $x\in [a,b]$ such that $f(x)=f(x+{\frac {1}{n}})$.

History

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

Statement of the theorem

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

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 $f(x)$ is continuous on some interval $I=[a,b]$ with the condition that $f(a)=f(b)$, then there exists some $x\in [a,b]$ such that $f(x)=f(x+1/2)$.

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

See also

• Intermediate value theorem
• Borsuk–Ulam theorem
• Rolle's theorem