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Q: Motivation for uniform continuity 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. A: 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