In mathematical analysis, the universal chord theorem states that if a function f is continuous on [a,b] and satisfies
f
(
a
)
=
f
(
b
)
{\displaystyle f(a)=f(b)}
, then for every natural number
n
{\displaystyle n}
, there exists some
x
∈
[
a
,
b
]
{\displaystyle x\in [a,b]}
such that
f
(
x
)
=
f...
Let $f \in C[0,1]$ and $f(0)=f(1)$.
How do we prove $\exists a \in [0,1/2]$ such that $f(a)=f(a+1/2)$?
In fact, for every positive integer $n$, there is some $a$, such that $f(a) = f(a+\frac{1}{n})$.
For any other non-zero real $r$ (i.e not of the form $\frac{1}{n}$), there is a continuous fu...
