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Math Help - Continuous Functions

  1. #1
    Senior Member
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    Continuous Functions

    I'm having trouble with this question

    Let\ f\ be\ continuous\ on\ \Re\ with\ \lim{x \rightarrow \inf} = \lim{x \rightarrow -\inf} = 0

    1)\ Give\ an\ example\ of such\ a\ function\ which\ has both\ a\ maximum\ value\ and\ a\ minimum\ value.

    2)\ Given\ an\ example\ of such\ a\ function\ which\ has\ a minimum\ value\ but\ no\ maximum\ value.

    3)\ Show\ that\ if\ there\ is\ a number\ \xi\ such\ that\ f(\xi)>0 then\ f\ attains\ a\ maximum\ value\ on\ \Re.
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  2. #2
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    1) The zero function has a maximum and minimum at every point. If this is cheating, just take the zero function and give it two "spikes", one above the x axis and one below it.

    2) -1\over 1+x^2

    3) Let f(\xi) = \zeta. Chuse an R so large that for all |x|>R, |f(x)| < \zeta. Consequently, we must have |\xi|\le R. Since f is continuous, it achieves a maximum on the compact interval [-R,R]. The maximum must be at least \zeta, so since outside of [-R,R], it is less than \zeta, the maximum it achieves on [-R,R] is the maximum on all of \mathbb{R}.
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