Let $f$ a continuous function on $\mathbb R$ such that $f(0)=f(2)$.

Question

Let $f$ a continuous function on $\mathbb R$ such that $f(0)=f(2)$. Answer by true or false. There exists $\alpha\in[0,1]$ such that $f(\alpha)=f(\alpha+1)$.

I think that it's wrong but I'm not able to find a counter exemple.

score 3 · Answer 1 · answered Nov 19 '14 at 15:42

3

Hint:

consider $g(x)=f(x)-f(x+1)$, note we always have $g(0)=-g(1)$. Discuss the value of $g(0),g(1)$. Also remember intermediate value theorem.

answered Nov 19 '14 at 15:42

John

13,738

This is certainly the way to go. One should note that $g:[0,1]\rightarrow \mathbb{R}$ – DrkVenom Nov 19 '14 at 15:45

Let $f$ a continuous function on $\mathbb R$ such that $f(0)=f(2)$.

1 Answers1