1. ## Functions Question

$\displaystyle Suppose\ that\ f\ is\ a\ continuous$ $\displaystyle function\ on\ the\ interval\ [0,1]\ and\ that\ f(0)=f(1).$

$\displaystyle 1)\ Show\ that\ f(a)=f(a+ \frac{1}{2})$ $\displaystyle for\ some\ a \in [0,\frac{1}{2}].$

$\displaystyle 2)\ If\ n\ is\ an\ integer\ greater\ than\ 2,\ show\ that$
$\displaystyle f(a)=f(a+ \frac{1}{n})\ for\ some\ a \in [0, 1- \frac{1}{n}].$

2. Originally Posted by acevipa
$\displaystyle Suppose\ that\ f\ is\ a\ continuous$ $\displaystyle function\ on\ the\ interval\ [0,1]\ and\ that\ f(0)=f(1).$

$\displaystyle 1)\ Show\ that\ f(a)=f(a+ \frac{1}{2})$ $\displaystyle for\ some\ a \in [0,\frac{1}{2}].$

Define $\displaystyle h(x):=f(x)-f(x+1/2)$ , and prove that either $\displaystyle h(0)=h(1/2)=0$ and then we're done, or else $\displaystyle h(0)h(1/2)<0$ and now apply the intermediate value theorem for h (why is it possible?) . Use the same procedure with the following question.

Tonio

$\displaystyle 2)\ If\ n\ is\ an\ integer\ greater\ than\ 2,\ show\ that$
$\displaystyle f(a)=f(a+ \frac{1}{n})\ for\ some\ a \in [0, 1- \frac{1}{n}].$
.

3. Originally Posted by tonio
.
I tried doing question 1 as you said using the interval $\displaystyle [0, \frac{1}{2}]$, but still don't know how to do it?

4. Originally Posted by acevipa
I tried doing question 1 as you said using the interval $\displaystyle [0, \frac{1}{2}]$, but still don't know how to do it?

What isn't clear in my answer? Where are you stuck?

Tonio