0

Suppose that $f$ is a continuous function on $[0,2]$ such that $f(0)=f(2)$.We have to show that there is a real number $c$ in the interval $[1,2]$, such that $f(c)=f(c-1)$.

I am completely lost on this question. I have tried fiddling with Rolles theorem and the Mean Value Theorem but it hasn't worked.

Sawarnik
  • 7,404

2 Answers2

4

Let $$h(x)=f(x)-f(x-1)$$ then $h$ is continuous on the interval $[1,2]$ and $h(1)h(2)=-(f(1)-f(0))^2\le0$ so by the intermediate value theorem there's $c\in[1,2]$ such that $h(c)=0$ which means $f(c)=f(c-1)$.

4

Hint Consider the function $g(x)=f(x)-f(x-1)$, and the fact that it is continuous. What values does it take at $x=1$ and $x=2$ ? Consider applying Intermediate Value theorem of continuous functions.

r9m
  • 18,208
  • Right! How did you hit upon this idea? – Sawarnik Mar 16 '14 at 07:49
  • @Sawarnik I think there are nice explanations of the intuition behind it in http://math.stackexchange.com/questions/612462/intuition-for-the-universal-chord-theorem – r9m Mar 16 '14 at 07:54