What do uniformly continuous functions look like?
My intuitive understanding of a uniformly continuous function is that it doesn't accelerate without bound and then stay there as the function itself increases (or decreases) without bound. So $x^2$ isn't uniformly continuous, because it gets faster and faster and faster and never slows down, but $\sqrt[3]{x}$ is, because despite being vertical $x=0$ it then slows down again.
Given any $\def\nn{\mathbb{N}}$$\def\rr{\mathbb{R}}$$n \in \nn$, any closed bounded subset of $\rr^n$ is compact.
Given any continuous function $f$ on a compact subset $S$ of $\rr^n$, we can prove that $f$ is uniformly continuous on $S$.
Thus any non-uniformly continuous function must have no (continuous) extension to a compact domain.
Two examples:
Let $f : \rr_{\ne 0} \to [0,1]$ such that $f(x) = \sin(\frac{1}{x})$ for any $x \in \rr_{\ne 0}$. Then $f$ is continuous on $\rr_{\ne 0}$ but not uniformly continuous. (The hole at $0$ cannot be patched.)
Let $f : \rr \to \rr$ such that $f(x) = x \sin(x)$ for any $x \in \rr$. Then $f$ is continuous on $\rr$ but not uniformly continuous. (An unbounded domain is never compact.)
Of course, some uniformly continuous functions do not have a domain that can be extended to a compact one, such as a step function.