1. ## question about continuous function.

hello,

i got continuous function on $[0,1]$ such that $f(0) = f(1)$ i need to prove that exist "a" such that $f(a) = f(a+0.5)$
$a\in \left[0,\frac{1}{2}\right]$

your help would be much appreciated!

2. ## Re: question about continuous function.

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

3. ## 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 $g(t)=f(t)-f(t+0.5)$.

4. ## Thanks

I knew it was something around the intermediate value theory
After reading all the article i found simmilar problem and it's solution.