Darboux property

Question

Possible Duplicate:
Universal Chord Theorem

$f: [0,1] \rightarrow \mathbb{R} $ is continuous and $ f(0) = f(1)$ prove that $\exists_{x_{0}\in[0,1]} $ such that $x_{0} + \frac{1}{2} \in [0,1]$ and $ f(x_{0})=f(x_{0}+\frac{1}{2})$

I know that have to use property Darboux but dont know how.

Answer 1

Consider the function $g(x)=f(x)-f(x+\frac12)$ on $[0,\frac12]$.

You want to show that $g(x)=0$ for some $x\in [0,\frac12]$.

Use the intermediate value theorem.