“Universal chord theorem”的意思、由来-开放百科全书

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Universal chord theorem

释义

History
Statement of the theorem
Case of n = 2
Proof of n = 2
Proof of general case
See also
References

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]

History

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

Statement of the theorem

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

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

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

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, g is continuous. Furthermore, . It follows that there exists integers such that

The intermediate value theorems gives us c such that and the theorem follows.

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. ^{{cite journal|last1=Oxtoby|first1=J.C.|title=Horizontal Chord Theorems|journal=The American Mathematical Monthly|date=May 1978|volume=79|pages=468–475|doi=10.2307/2317564}}

1 : Mathematical theorems

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