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Math Help - 2 more fun problems, please help!

  1. #1
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    2 more fun problems, please help!

    Dear math helpers, thank you for all your help thus far. These are two more problems I was having trouble with and would appreciate feedback. Thank you in advance



    1.Decide whether the following statements are true or false. Justify your
    answers: proof or counterexample.
    (a) Every continuous function f : [0, 1) → R which is bounded
    takes on its maximum.
    (b) There exists a function f : [−1, 1] → [−1, 1] with no x ∈ [−1, 1]
    satisfying f (x) = x.


    2.Suppose that the function f is differentiable at a. Prove (without
    quoting a theorem) that f 2 is differentiable at a.
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  2. #2
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    Quote Originally Posted by Swamifez View Post
    (b) There exists a function f : [−1, 1] → [−1, 1] with no x ∈ [−1, 1]
    satisfying f (x) = x.
    It depends whether f is continous or not. If it is then there is no such function, this is a "Brouwer Fixed Point (I think)"
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  3. #3
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    The Perfect Hacker, thank you for your post before. assume f is contimous in all my problems. Thanks again
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  4. #4
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    Quote Originally Posted by Swamifez View Post
    1.Decide whether the following statements are true or false. Justify your
    answers: proof or counterexample.
    (a) Every continuous function f : [0, 1) → R which is bounded
    takes on its maximum.
    The limit \lim_{x\to 1^-}f(x) exists for f is continous and bounded. Let L be that limit.

    Redefine the function as,
    g(x)=\left\{ \begin{array}{c}f(x) \mbox{ for }0\leq x<1\\ L \mbox{ for }x=1 \end{array}\right\}
    This new function is countinous on the closed interval [0,1] EVT tells us it has a maximum value.
    Last edited by ThePerfectHacker; November 16th 2006 at 06:33 AM.
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