What does "bounded away from zero" actually mean?

If a set $S \subset \mathbb R$ is bounded away from zero, it means that there exists $m > 0$ such that $|x| > m$ 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 $m > 0$ such that $|f(x)| > m$ 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.

Not to start an argument, but to give the way that I've heard this phrase used:

$f$ is bounded away from zero if there is $\varepsilon > 0$ such that the range of $f$ does not meet $(-\varepsilon, \varepsilon)$, or equivalently, there is some open set containing zero which the range of $f$ does not meet.

In this sense, the function $f(z) = 1/z$ is bounded on $0 < z < 1$, as the range of $f$ does not meet, say, $(-\frac{1}{2},\frac{1}{2})$.