TITLE: Example of continuous function on closed and unbounded set in $R$ with no maximum
QUESTION [5 upvotes]: What is an example of a continuous function on a closed and unbounded set with no maximum? Is $f(x)=x^3$ a correct example?

REPLY [4 votes]: Your example is correct. Simpler might be $f(x) = x$ and maybe more interesting $f(x) = x^{37} + x^{10} +1$, but not $f(x) = - x^2$ (why?)
Some other examples:

$f(x) = -e^{x}$: bounded above by $0$ but has no maximum.
$f(x) = -e^{-x^2}$: bounded above, assumes its minimum $-1$ at $0$ is bounded above by $0$ but has no maximum.
$f(x) = \arctan{x}$: bounded above and below by $\pm \frac{\pi}{2}$ but has neither maximum nor a minimum.

See the plots by Wolfram|Alpha.