## Universal Chord Theorem

Let fC[0,1] and f(0)=f(1).
How do we prove a[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+1n).
For any other non-zero real r (i.e not of the form 1n), there is a continuous function fC[0,1], such that f(0)=f(1) and f(a)f(a+r) for any a.