TITLE: Visualizing Uniform Continuity
QUESTION [14 upvotes]: Given a colloquial definition of uniform continuity as $f(x)$ and $f(y)$ can be made to be arbitrarily close when $x$ and $y$ are sufficiently close, and the distance between $x$ and $y$ is independent of $x$ and $y$.
I'm not really sure how to picture a uniformly continuous function in my head. I showed that if the derivative of a function is bounded, then it will be uniform continuous. (I had trouble with the converse though.) Thinking along the lines that I need to bound the change in $f$.
How do you visualize uniform continuity?

REPLY [5 votes]: Here is a visualization of uniform continuity: 
Let $\gamma$ be the graph of a function $f\colon\  {\mathbb R}\to{\mathbb R}$ and let an $\epsilon>0$ be given. The $\epsilon$-tube with soul $\gamma$ is the set $T_\epsilon(\gamma):=\{(x,y)\in{\mathbb R}^2\ |\ |y- f(x)|<\epsilon\}$. 
Together with $f$ we consider its translates $f_a$ defined by $f_a(x):=f(x-a)$. Let $\gamma_a$ be the graph of $f_a$.
The given function $f$ is uniformly continuous on ${\mathbb R}$ if, given an $\epsilon>0$, there is a $\delta>0$ such all graphs $\gamma_a$ for $|a|<\delta$ fit into the $\epsilon$-tube with soul $\gamma$:
$$\gamma_a\subset T_\epsilon(\gamma)\qquad \bigl(|a|<\delta\bigr)\ .$$