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Find a counterexample for the following statement.

"Let $f : [1, 6] \to \mathbb{R}$ be a continuous function such that $f(1) = f(6)$. Show that there exists $p \in [1, 6]$ such that $f(p) = f(p + 3)$."

If the problem would have been $f(p) = f(p + 3.5)$, then we can just apply IVP on $g(x) = f(x) - f(x+3.5)$. However, for $f(p) = f(p + 3)$, this seems to be challenging to prove. I am not getting a counterexample, either. Please help.

Summation
  • 350
  • The link shows the Universal Chord Theorem, which says that the only lengths that are guaranteed to have a chord are $\frac 1n$ of the total length. As your chord is length $3$ out of $5$ the statement is not true. One of the answers shows how to construct a function where there is no such chord. I think in your last paragraph the $3.5$s should be $2.5$s, which are half the interval. – Ross Millikan Nov 24 '24 at 05:05
  • Boy is this a difficult category. Never ceases to amaze me. – suckling pig Nov 24 '24 at 05:17

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