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 .

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

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.

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.

See also
Intermediate value theorem
Borsuk–Ulam theorem
Rolle's theorem

References

Category:Mathematical theorems