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Math Help - Functions Question

  1. #1
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    Functions Question

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

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


    2)\ If\ n\ is\ an\ integer\ greater\ than\ 2,\ show\ that
    f(a)=f(a+ \frac{1}{n})\ for\ some\ a \in [0, 1- \frac{1}{n}].
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  2. #2
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    Quote Originally Posted by acevipa View Post
    Suppose\ that\ f\ is\ a\ continuous function\ on\ the\ interval\ [0,1]\ and\ that\ f(0)=f(1).

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


    Define h(x):=f(x)-f(x+1/2) , and prove that either h(0)=h(1/2)=0 and then we're done, or else 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


    2)\ If\ n\ is\ an\ integer\ greater\ than\ 2,\ show\ that
    f(a)=f(a+ \frac{1}{n})\ for\ some\ a \in [0, 1- \frac{1}{n}].
    .
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  3. #3
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    Quote Originally Posted by tonio View Post
    .
    I tried doing question 1 as you said using the interval [0, \frac{1}{2}], but still don't know how to do it?
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  4. #4
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    Quote Originally Posted by acevipa View Post
    I tried doing question 1 as you said using the interval [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
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