Motivation for uniform continuity

Question

What are some problems or theorems that motivate the distinction between continuity and uniform continuity? In particular, I would like:

a) A useful, appealing theorem that applies to uniformly continuous functions but not arbitrary continuous functions, and there is an easy example of a continuous (but not uniformly) function for which the conclusion of the theorem fails,

or

b) A natural, fun problem that can be solved using a uniformly continuous function and can't be solved using a function that is merely continuous.

The closest thing I currently have to what I'm looking for is a version of the Riemann-Lebesgue lemma: $\int_a^b \sin(Nx)f(x)dx \to 0$ as $N\to \infty$, if $f$ is uniformly continuous on $(a,b)$, but not if it's merely continuous. But my students do not yet know Fourier analysis, so they won't know why this is a useful or natural question to answer.

Answer 1

First, I think this is a great question for this site because it made me question how I teach uniform continuity in my own classes and sent me on an exploration for better ideas. So, thanks!

Now, to address the question. In the past, to motivate uniform continuity (after introducing it), I would use a challenging exercise about Cauchy sequences and an extension of a function's domain. I use Abbott's Understanding Analysis (2nd Ed.), and the exercise is 4.4.13 (Continuous Extension Theorem):

(a) Show that a uniformly continuous function preserves Cauchy sequences; that is, if $f:A\to\mathbb{R}$ is uniformly continuous and $(x_n)\subseteq A$ is a Cauchy sequence, then show $f(x_n)$ is a Cauchy sequence.

(b) Let $g$ be a continuous function on the open interval $(a,b)$. Prove that $g$ is uniformly continuous on $(a,b)$ if and only if it is possible to define values of $g(a)$ and $g(b)$ at the endpoints so that the extended function $g$ is continuous on $[a,b]$. (In the forward direction, first produce candidates for $g(a)$ and $g(b)$, and then show the extended $g$ is continuous.)

This is fine and should certainly appeal to us mathematicians. However, for the students, I'm sure this feels quite distant from what else they've seen about uniform continuity, when it's applied to specific functions (e.g. $x^2$ is uniformly continuous on $[0,1]$ but not $[0,\infty)$.) Moreover, when I teach Real Analysis, a significant course objective is to teach students how to think about (and, indeed, even care about) abstraction and generality. So, I worry about heavy cognitive load in this section: why would a student care about this new concept (uniform continuity) when the reason to do so is merely to prove this rather general theorem about arbitrary sequences and continuous functions, especially if the student has difficulty even understanding what the theorem states?!

So, I've developed an idea that I am suggesting to you and myself. (Yes, I haven't tried this yet, but I believe it will work well.) Use this as an occasion to look back to some results from precalculus and calculus (and intro to proofs), as well as what has been discussed in class about the relationship between $\mathbb{Q}$ and $\mathbb{R}$. Specifically, prove in several steps that the exponential function $f(x)=2^x$ can be defined on the domain $\mathbb{R}$ and is continuous on that domain.

The list below could be completed as an investigative project by the students (perhaps in teams) or together with you in class, or some combination thereof.

1. Use the "repeated multiplication" concept to explain why $2^x$ exists for $x\in \mathbb{N}$.
2. Use the idea that $2^{-x}=\frac{1}{2^x}$ to explain why $2^x$ exists for $x\in\mathbb{Z}$.
3. Use the Intermediate Value Theorem to explain why $2^{1/n}$ exists for $n\in\mathbb{N}$. (If $2^{1/n}=y$ then $y^n=2$, so consider the function $f(y)=y^n$.)
4. Explain why $2^x$ exists for $x\in\mathbb{Q}$.

So far, this has been pretty straightforward, and the IVT is the only necessary result from analysis. Here comes the new & fun stuff, which was partly inspired by the discussion in "Relations to the extension problem" on the Wikipedia page for "Uniform continuity": https://en.wikipedia.org/wiki/Uniform_continuity#Relations_with_the_extension_problem

5. Show that $f(x)=2^x$ is not uniformly continuous on the domain $\mathbb{Q}$. (Remember that we don't know yet whether $2^x$ exists for $x\notin\mathbb{Q}$.)
6. Let $I$ be a bounded interval. Show that $f$ is uniformly continuous on the domain $I\cap \mathbb{Q}$. (You can do this for this specific function, or prove a more general result.)
7. Show that $f$, restricted to $I$, preserves Cauchy sequences, because it is uniformly continuous on that domain. (Again, do this specifically or generally, depending on the audience.)
8. Explain why this means we can extend $f$ to the domain $I$ (not just $I\cap\mathbb{Q}$) in such a way that $f$ is continuous on $I\subseteq\mathbb{R}$. This will necessitate a discussion with students about the completeness of $\mathbb{R}$ and Cauchy sequences of rationals converging to irrationals, as well as continuity and limits of sequences. In other words, this should tie together many concepts from earlier in the course!
9. Explain why, since this can be done for any bounded interval $I$, there must be a unique way to extend $f$ to all of $\mathbb{R}$ and make it continuous on that domain.
10. If desired, share more general versions of some of these results. For example: If $X$ is a complete metric space and $f:S\to X$ is uniformly continuous for every $S\subseteq X$ that is bounded, then $f$ may be extended to a continuous function on the domain $X$.

Edit: I forgot to mention that step #7 is the one that truly addresses your question, specifically goal (a). A continuous function $f$ need not necessarily preserve Cauchy-ness of a sequence, but a uniformly continuous function does. The Math.SE link in the comments expands on this idea, but rather than just talk about that fact to the students, my answer here suggests using that idea within a specific (and, I hope, more compelling for them) example.