Having moved through the preceding compactness ideas, we now study extreme values on compact domains. The focus keyword for this lesson is extreme values 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[Extreme Value Theorem on Compact Metric Spaces] Let $X$ be compact and let $f:X\to\mathbb{R}$ be continuous. Then $f$ is bounded and attains its maximum and minimum. ::: :::proof Given that $X$ is compact and $f$ is continuous. To prove the theorem. The set $f(X)$ is compact in $\mathbb{R}$, hence closed and bounded. Therefore its supremum and infimum belong to $f(X)$. Thus $f$ attains maximum and minimum values. $\square$ ::: :::example[Failure Without Compactness] The function $f(x)=1/x$ on $(0,1)$ is continuous but unbounded. The function $g(x)=(x-1)/x$ on $(1,\infty)$ is bounded but does not attain $1$. ::: :::solved-problem If $A$ is compact and $B$ is nonempty, the distance from $A$ to $B$ is attained at some point of $A$. ::: :::solution The map $x\mapsto d(x,B)$ is continuous on compact $A$, so it attains its minimum. $\square$ ::: :::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 Compact Domains and Homeomorphisms] 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/compact-domains-and-homeomorphisms button-ghost: Previous Lesson | /functional-analysis/metric-spaces/compactness-and-its-consequences-in-metric-spaces/compactness-under-continuous-mappings :::