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Thread: Continuous Functions

  1. #1
    Senior Member
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    Feb 2008
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    Continuous Functions

    I'm having trouble with this question

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

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

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

    $\displaystyle 3)\ Show\ that\ if\ there\ is\ a$ $\displaystyle number\ \xi\ such\ that\ f(\xi)>0$ $\displaystyle then\ f\ attains\ a\ maximum\ value\ on\ \Re.$
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  2. #2
    Senior Member
    Joined
    Feb 2010
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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) $\displaystyle -1\over 1+x^2$

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