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 $\mathbb{R}$.
Best Answer
I like to think of the following fact: a $C^1$ function on $\mathbb{R}$ with bounded derivative is uniformly continuous. So in order for a function not to be uniformly continuous, there have to be places where its graph is "arbitrarily steep".
Because of the theorem that any continuous function on a compact set is uniformly continuous, this can only happen if you make the function steeper and steeper as you go off to $\infty$, either unboundedly (like $x^2$) or boundedly (like $\sin(x^2)$).
Edit: As pointed out, the converse is false: a function with unbounded derivative can still be uniformly continuous. $\sin(x^4)/(1+x^2)$ is such an example, I believe. So this may not be great intuition after all.