Find a continuous and bounded function on $[0,\infty)$ where the extreme value theorem does not hold.

Question

Find a continuous and bounded function on $[0,\infty)$ where the extreme value theorem does not hold.

I think I may need to use the Balzono-Weierstrass Theorem for this problem but I'm having trouble getting started.

Answer 1

You don't need to use a theorem, rather you need to come up with an explicit example of a continuous function on $[0,\infty)$ which fails to attain its maximum or its minimum (or both).

Since the extreme value theorem holds on any closed and bounded interval, your example needs to make use of the "extra room" in $[0,\infty)$. I suggest trying a continuous increasing function $f(x)$ such that $L=\lim_{x\to\infty}f(x)$ exists but $f(x)<L$ for all $x\geq 0$.

Answer 2

Take a function which approaches a limit from below as it goes to $\infty$. It will not attain its maximum.

Answer 3

f := [0, $\infty$) $\rightarrow$ [0, 1)

f(x) = $x^2 \over{x^2+1}$

Answer 4

$x/(|x|+1)$ is continuous and bounded and does not attain a maximum