question about continuous function.

hello,

i got continuous function on $\displaystyle [0,1]$ such that $\displaystyle f(0) = f(1)$ i need to prove that exist "a" such that $\displaystyle f(a) = f(a+0.5)$

$\displaystyle a\in \left[0,\frac{1}{2}\right]$

your help would be much appreciated!

Re: question about continuous function.

The idea is described in the Wikipedia article about the intermediate value theorem.

Re: question about continuous function.

Emakarov is correct. Look at the example at the bottom of the article concerning real world applications. Your problem is just a slight variation: consider the function $\displaystyle g(t)=f(t)-f(t+0.5)$.