I understand the formal definition of uniform continuity of a function, and how it is different from standard continuity.
My question is: Is there an intuitive way to classify a function on $\mathbb{R}$ as being uniformly continuous, just like there is for regular continuity? I mean, that for a "nice" function $f:\mathbb{R} \to \mathbb{R}$, it is usually easy to tell if it is continuous on an interval by looking at or thinking of the graph of the function on the interval, or on all of $\mathbb{R}$. Can I do the same for uniform continuity? Can I tell that a function is uniformly continuous just by what it looks like? Ideally, this intuition would fit with the Heine-Cantor theorem for compact sets on $\mathbb{R}$.