Having moved through the preceding compactness ideas, we now study uniform continuity on compact domains. The focus keyword for this lesson is uniform continuity on compact domains. This page records the definitions, theorems, proofs, examples, and practice items needed for undergraduate work in compactness and its consequences in metric spaces. :::theorem[Continuous Functions on Compact Spaces Are Uniformly Continuous] Let $X$ be compact and let $f:X\to Y$ be continuous. Then $f$ is uniformly continuous. ::: :::proof Given that $X$ is compact and $f$ is continuous. To prove uniform continuity. For each point, continuity gives a local radius. These local balls form an open cover. A Lebesgue number for this cover supplies one radius that works for the whole space. $\square$ ::: :::corollary Every continuous function on a closed bounded interval $[a,b]$ is uniformly continuous. ::: :::theorem[Dini-Type Uniform Convergence] On a compact metric space, a monotone sequence of continuous real-valued functions converging pointwise to a continuous limit converges uniformly. ::: :::example[Compactness Cannot Be Dropped] The sequence $f_n(x)=1/(1+nx)$ on $(0,1)$ decreases pointwise to $0$ but not uniformly. ::: :::exercise [1] State the main definition or theorem from this lesson. [2] Give one example where the hypothesis is used. [3] Explain one common mistake related to this topic. ::: :::answer [1] See the first formal block of the lesson. [2] The examples in the lesson provide standard choices. [3] A common mistake is to apply a Euclidean compactness rule in an arbitrary metric space. ::: :::faq[Frequently Asked Questions] Q: Why is the metric important? A: Compactness, boundedness, convergence, and openness are all interpreted using the chosen metric. Q: What should students check first? A: Identify the space, the metric, and the exact compactness criterion being used. ::: :::call-to-action[Continue to Local Compactness] subtitle: Review the definitions, proofs, and examples before moving to the next compactness idea. button: Next Lesson | /functional-analysis/metric-spaces/compactness-and-its-consequences-in-metric-spaces/local-compactness button-ghost: Previous Lesson | /functional-analysis/metric-spaces/compactness-and-its-consequences-in-metric-spaces/compact-domains-and-homeomorphisms :::