Universal chord theorem

In mathematical analysis, the universal chord theorem states that if a function f is continuous on [a,b] and satisfies , then for every natural number , there exists some such that .[1]

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.

History

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

Statement of the theorem

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Let   denote the chord set of the function f. If f is a continuous function and  , then   for all natural numbers n. [3]

Case of n = 2

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The case when n = 2 can be considered an application of the Borsuk–Ulam theorem to the real line. It says that if   is continuous on some interval   with the condition that  , then there exists some   such that  .

In less generality, if   is continuous and  , then there exists   that satisfies  .

Proof of n = 2

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Consider the function   defined by  . Being the sum of two continuous functions,   is continuous,  . It follows that   and by applying the intermediate value theorem, there exists   such that  , so that  . Which concludes the proof of the theorem for  

Proof of general case

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The proof of the theorem in the general case is very similar to the proof for   Let   be a non negative integer, and consider the function   defined by  . Being the sum of two continuous functions,   is continuous. Furthermore,  . It follows that there exists integers   such that   The intermediate value theorems gives us c such that   and the theorem follows.

See also

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References

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  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.