Describe all functions $f: [a, b] \rightarrow \mathbb{R}$ that reach their maximum and its minimum over all subset $A \subset{ [a, b]}$

Question

My question:

Describe all functions $f: [a, b] \rightarrow \mathbb{R}$ that reach their maximum and its minimum over all subset $A \subset{[a, b]}$

As a description of the functions, what I have tried is to see that the function does not necessarily have to be continuous, we can define one that obtains it and that is not continuous, for this reason the number of jumps that the function gives would not necessarily have to be countable, but I don't know if there is a stronger characteristic that describes all of them in general.

Similarly, in this case it seems to me that not all uniformly continuous functions do not necessarily have to fulfill them.

But I don't really know what would be a way to describe all the functions.

Answer 1

Outline:

• The function $f$ has your property iff $f((a,b))$ has the property that every subset of it contains a maximal and a minimal element.
• An infinite subset of $\mathbb{R}$ contains a strictly monotone sequence.
• A strictly monotone sequence contains either no maximal or no minimal element.
• Hence, $f((a,b))$ is necessarily finite and that is obviously sufficient as well.