Why does a continuous function on an open interval achieve either an absolute minimum or max but not both?

Question

In my analysis textbook, there is an exercise that states:

Exercise. Let $f$ be a continuous function defined on the interval $(0,1)$ to the real numbers, such that $\lim_{x\to 0} f(x)=0$ and $\lim_{x\to 1} f(x) = 0$.

Show $f$ achieves either an absolute minimum or an absolute maximum on $(0,1)$ but perhaps not both.

My question is, why is it 'perhaps not both'? For example, if I defined a function on $(0,2\pi)$ and let $f(x) = \sin(x)$, then wouldn't the function have both an absolute maximum and minimum?

Answer 1

That “perhaps not both” means that sometimes such a function has a maximum but not a minimum or a minimum but no a maximum. Take $f(x)=x-x^2$, for instance. It has a maximum, but not a minimum.

Answer 2

A continuous function on an open interval with the given limits could be extended to a continuous interval on the closed interval with $f(0)=f(1)=0$

Thus for the extended function you have to have both absolute max or absolute minimum on the closed interval.

Now in case that one of them happens at the endpoint the other will be inside unless the function is a constant.