What does it mean to say that a function assumes its supremum or infimum?

Question

I have to find the following 3 things:

1. A function on interval $[0, 1]$ that does not assume its supremum.

2. A continuous function on $[1, +\infty)$ that does not assume its infimum.

3. A continuous function of $(0, 1)$ that does not assume its supremum or infimum.

I know what supremum and infimum are but I don't understand what I'm being asked to do. What does "assume" mean? Thanks in advance.

Answer 1

When we say a function $f: S \longrightarrow \mathbb{R}$ assumes its supremum, it means there is some $s \in S$ with $f(s)=\sup\{f(x)\,|\,x \in S\}$. In other words, $f$ has a maximum.

Similarly for the infimum.

For instance, $f:(0,+\infty)\longrightarrow \mathbb{R}$ given by $f(x)=\frac{1}{x}$ does not assume its supremum, but $g:\mathbb{R}\longrightarrow \mathbb{R}$ given by $g(x)=-x^2$ does, at $x=0$.

Answer 2

A function on interval [0, 1] that does not assume its supremum.

This is the hardest of the three. A hint, every continuous function achieves its supremum and its infimum. (extreme value theorem) Therefore, this is not a continuous function. Right where the function is about to hit its maximum, create a discontinuity.

regarding 2. the interval is probably $[0,\infty)$ i.e. open and not closed at that end. Check it. Anyway, think of a horizontal asymptote.

3. Is the easiest of these. Any monotonic function will do.