Saturday, 5 October 2019

real analysis - A problem related to intermediate value property of continuous function.










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 nN 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
$$f(x)=\begin{cases}x&\mathrm{if\ }x\le \frac13\\
1-2x &\mathrm{if\ } \frac13x-1 &\mathrm{if\ }x\ge\frac23
\end{cases}$$


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