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**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

$\displaystyle 2)\ If\ n\ is\ an\ integer\ greater\ than\ 2,\ show\ that$

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