This is actually not my question, it was asked yesterday by user176744 in this link
$[0,n]$ continuous function problem
and I feel as if it didn't get enough attention. I am also interested in a solution to this problem. As shown in the link I did manage to prove it, but only after assuming many things which were not stated in the question.
Can anyone think of a proof without extra assumptions or a counter example that will disprove the statement?
the question:
Let $n \in \mathbb N$ and $f: [0,n] \to \mathbb R$ such that $f$ is continuous on $[0,n]$ and $f(0)=f(n)$.
Prove that there are $x_1,x_2 \in [0,n]$ such that $x_1-x_2=1$ and $f(x_1)=f(x_2)$
You can see my solution in the link, but as said, too many assumptions. I am looking for a better solution or a counter example.
