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
n
, there exists some
x
∈
[
a
,
b
]
such that
f
(
x
)
=
f
(
x
+
1
n
)
.
The theorem was published by Paul Lévy in 1934 as a generalization of Rolle's Theorem.
Let
H
(
f
)
=
{
h
∈
[
0
,
1
]
:
f
(
x
)
=
f
(
x
+
h
)
for some
x
}
denote the chord set of the function f. If f is a continuous function and
h
∈
H
(
f
)
, then
h
n
∈
H
(
f
)
for all natural numbers n.
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 some interval
I
=
[
a
,
b
]
with the condition that
f
(
a
)
=
f
(
b
)
, then there exists some
x
∈
[
a
,
b
]
such that
f
(
x
)
=
f
(
x
+
1
/
2
)
.
In less generality, if
f
:
[
0
,
1
]
→
R
is continuous and
f
(
0
)
=
f
(
1
)
, then there exists
x
∈
[
0
,
1
2
]
that satisfies
f
(
x
)
=
f
(
x
+
1
/
2
)
.
