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Guarantees chords of length 1/n exist for functions satisfying certain conditions From Wikipedia, the free encyclopedia
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]
The theorem was published by Paul Lévy in 1934 as a generalization of Rolle's Theorem.[2]
Let denote the chord set of the function f. If f is a continuous function and , then for all natural numbers n. [3]
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 .
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
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.
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