Let f:[0,1]→R be a real valued continuous function satisfying f(0)=f(1). Then using intermediate value theorem we know for every n∈N there exist two point a,b∈[0,1] at a distance 1/n satisfying f(a)=f(b).
Now my question is, for every r∈[0,1] is it possible to find two points a,b∈[0,1] at a distance r, satisfying f(a)=f(b) provided f:[0,1]→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=23 and
$$f(x)=\begin{cases}x&\mathrm{if\ }x\le \frac13\\
1-2x &\mathrm{if\ } \frac13
\end{cases}$$
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