Coming up with a counter example - calculus

Question

I have to come up with a counter example for the following statement:

Let $f$ be a function $f: [0,\infty)\longrightarrow R$, continuous and bounded. Prove that it receives either a minimum or a maximum (or both)

Everything I tried seems to always be lacking one of the conditions, for example $\sin(1/x)$ is not continuous at $0$, $x\sin(x)$ is unbounded, $\sin(x) + 1/x$ does have a minimum

Any help would be appreciated.

Answer 1

There are two ways that a function can fail to achieve a maximum:

1. It can increase to $\infty$
2. It can approach the maximum, without quite reaching it, as $1-\frac1x$ increases toward $1$ without getting there

Since the function you are looking for is bounded, (1) is ruled out. It must be (2).

Similarly, if the function fails to achieve a minimum, it must approach its greatest lower bound closer and closer without getting there.

If the function approached one of its bounds and stayed close to that bound, the way $1-\frac 1x$ does, the function wouldn't get arbitrarily close to the other bound. So it can't be monotonic. It must wiggle back and forth between the two bounds, getting close to the upper bound, then close to the lower bound, then closer to the upper bound, then closer to the lower bound.

Does that help?

Answer 2

You can take, for instance, $f(x)=\sin(x)\arctan(x)$. Its supremum and its infimum are $\pm\pi/2$, but none of them is reached. Besides, it is continuous and bounded (for each $x\in[0,\infty)$, $|f(x)|<\pi/2$).