Universal chord theorem explained

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

, there exists some

x\in[a,b]

such that

f(x)=f\left(x+

	b-a
	n

\right)

.^[1]

History

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

Statement of the theorem

Let

H(f)=\{h\in[0,+infty):f(x)=f(x+h)forsomex\}

denote the chord set of the function f. If f is a continuous function and

h\inH(f)

, then

	h
	n

\inH(f)

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

f(x)

is continuous on someinterval

I=[a,b]

with the condition that

f(a)=f(b)

, then there exists some

x\in[a,b]

such that

f(x)=f\left(x+

	b-a
	2

\right)

In less generality, if

f:[0,1] → \R

is continuous and

f(0)=f(1)

, then there exists

x\in\left[0,

	1
	2

\right]

that satisfies

f(x)=f(x+1/2)

Proof of n = 2

Consider the function

g:\left[a,\dfrac{b+a}{2}\right]\toR

defined by

g(x)=f\left(x+\dfrac{b-a}{2}\right)-f(x)

. Being the sum of two continuous functions,

is continuous,

g(a)+g(\dfrac{b+a}{2})=f(b)-f(a)=0

. It follows that

g(a) ⋅ g(\dfrac{b+a}{2})\le0

and by applying the intermediate value theorem, there exists

c\in\left[a,\dfrac{b+a}{2}\right]

such that

g(c)=0

, so that

f(c)=f\left(c+\dfrac{b-a}{2}\right)

. Which concludes the proof of the theorem for

n=2

Proof of general case

The proof of the theorem in the general case is very similar to the proof for

n=2

Let

be a non negative integer, and consider the function

g:\left[a,b-\dfrac{b-a}{n}\right]\toR

defined by

g(x)=f\left(x+\dfrac{b-a}{n}\right)-f(x)

. Being the sum of two continuous functions,

is continuous. Furthermore,

	n-1
\sum
	k=0

g\left(a+k ⋅ \dfrac{b-a}{n}\right)=0

. It follows that there exists integers

i,j

such that

g\left(a+i ⋅ \dfrac{b-a}{n}\right)\le0\leg\left(a+j ⋅ \dfrac{b-a}{n}\right)

The intermediate value theorems gives us c such that

g(c)=0

and the theorem follows.

References

Rosenbaum, J. T. (May, 1971) The American Mathematical Monthly, Vol. 78, No. 5, pp. 509–513
[Paul Lévy (mathematician)|Paul Levy]
Oxtoby. J.C.. Horizontal Chord Theorems. The American Mathematical Monthly. May 1978. 79. 468–475. 10.2307/2317564.

Universal chord theorem explained

History

Statement of the theorem

Case of n = 2

Proof of n = 2

Proof of general case

See also

References