My question is rather a semantic one.
I am wondering what uniformity in different contexts in general means. I know what the definition of uniform continuity is. Has uniformity in general something to do with this notion (i.e., $\delta$-$\varepsilon$-criterion)?
For example, in a paper I read that the inequality on $x \in (0,1]$ $$ h(x) \leq C x^{-2} $$ means that $h(x)$ blows up uniformly (where $C \in \mathbb{R}$ is constant). What does uniformity mean here? The function $x^{-2}$ on the r.h.s. is not uniformly continuous on $(0,1]$. Or it has something to do with the constant $C$ on the r.h.s. since $C$ remains constant, independent of $x \in (0,1]$?
"Uniform" usually means roughly "the same everywhere". In the case of uniform continuity, given some $\epsilon$, the same $\delta$ works everywhere (or, at least, there is some $\delta$ that works everywhere). It's the continuity that says $\epsilon$-$\delta$ is involved, not the uniformity.
Similarly, a (usually infinite) collection of sequences (including sequences of real functions) is said to converge uniformly to some limit if for any $\epsilon$, you can find an $N$ that works for all sequences simultaneously.
As another example, uniform probability means any possible outcome is equally probable. I'm sure there are others.