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Possible Duplicate:
Universal Chord Theorem

Let $f:[0,1] \to R$ be a real valued continuous function satisfying $f(0)=f(1)$. Then using intermediate value theorem we know for every $n \in N$ there exist two point $a,b \in [0,1]$ at a distance $1/n$ satisfying $f(a)=f(b)$.

Now my question is, for every $r\in [0,1]$ is it possible to find two points $a,b\in [0,1]$ at a distance $r$, satisfying $f(a)=f(b)$ provided $f:[0,1] \to R$ be a real valued continuous function satisfying $f(0)=f(1)$.

as there is a counterexample for $r>1/2$, please consider the case when $r<1/2$.

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    @martin sorry martin but i was not aware of the universal chord theorem. And thanks for your references.2012-09-20

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Hint: Consider $r=\frac23$ and $f(x)=\begin{cases}x&\mathrm{if\ }x\le \frac13\\ 1-2x &\mathrm{if\ } \frac13

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    thank you for your answer.I have one doubt. For r< 1/2 will we be able to produce counterexample.2012-09-20