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