---- > [!theorem] Theorem. ([[extreme value theorem]]) > Let $f:X \to Y$ be a [[continuous]] function between a [[compact]] [[topological space]] $X$ and an [[poset|totally ordered set]] $Y$ in the [[order topology]]. Then $f$ attains a maximum and minimum value on $X$. > [!specialization] Specializing. > The familiar [[extreme value theorem]] from calculus is obtained when we take $X$ to be a [[closed interval]] in $\mathbb{R}$ and $Y=\mathbb{R}$. > [!proof]- Proof. ([[extreme value theorem]]) > > As usual, we implicitly employ [[compactness characterization for subspaces]] throughout. > Since $X$ is [[compact]] and $f$ [[continuous]], the set $A=f(X)$ is [[compact]]. Suppose $A$ does not contain a maximum. Take the covering $A=\{ (-\infty, a) : a \in A\}$ with finite subcovering $A=\{ (-\infty, a_{1}), \dots, (-\infty, a_{n}) \}.$ If $a_{i}$ is the largest of $a_{1},\dots,a_{n}$, then $a_{i} \notin A$, contradicting the assumption that the collection covers $A$. So $A$ must contain a maximum. > Same reasoning applies for minimum. ---- #### ----- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as Tag > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```