
Continuous Functions
I'm having trouble with this question
$\displaystyle Let\ f\ be\ continuous\ on\ \Re\ with\ \lim{x \rightarrow \inf} = \lim{x \rightarrow \inf} = 0$
$\displaystyle 1)\ Give\ an\ example\ of$ $\displaystyle such\ a\ function\ which\ has$ $\displaystyle both\ a\ maximum\ value\ and\ a\ minimum\ value.$
$\displaystyle 2)\ Given\ an\ example\ of$ $\displaystyle such\ a\ function\ which\ has\ a$ $\displaystyle minimum\ value\ but\ no\ maximum\ value.$
$\displaystyle 3)\ Show\ that\ if\ there\ is\ a$ $\displaystyle number\ \xi\ such\ that\ f(\xi)>0$ $\displaystyle then\ f\ attains\ a\ maximum\ value\ on\ \Re.$

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 $\displaystyle \mathbb{R}$.