What does "bounded away from zero" actually mean? For example, is $f(z) = 1/z$, on the set $0&lt;z&lt;1$ "bounded away from zero"?
If a set $S \subset \mathbb R$ is bounded away from zero, it means that there exists $m &gt; 0$ such that $|x| &gt; m$ for all $x \in S$.

If a function $f$ is bounded away from zero, it means that its range is bounded away from zero: there exists $m &gt; 0$ such that $|f(x)| &gt; m$ for all $x$.

Edited to clarify: When we say a set is bounded away from zero, we are not saying that away from zero, it is bounded. What would that even mean? We are saying that its distance from zero is bounded below by a strictly positive number. I see now that this is not self-evident, but that's what it means.
i emphasize, that bounded away from zero means correctly, that:

For all $\epsilon&gt;0$: $f([\epsilon,1])$ is bounded