1. ## Continuous Functions

I'm having trouble with this question

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

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

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

$3)\ Show\ that\ if\ there\ is\ a$ $number\ \xi\ such\ that\ f(\xi)>0$ $then\ f\ attains\ a\ maximum\ value\ on\ \Re.$

2. 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 $\mathbb{R}$.