Theorem: Continuous Functions Mapping Compact Domains to Real Numbers Take Maximum and Minimum Values on these Domains 

Let $X$ be a metric spaces, $D\subset X$ be a compact subset and $f:D\mapsto \mathbb R$ a continuous function mapping the domain $D$ to the real numbers $\mathbb R$. Then there are points \(p,q\in X\), for which the function $f$ takes the maximum and the minimum values of the image $f(D)$, formally $$\exists p,q\in X:~f(p)=\max(f(D)),\quad f(q)=\min(f(D)).$$

| | | | | created: 2017-03-11 21:19:38 | modified: 2017-03-12 13:44:50 | by: bookofproofs | references: [582]

1.Proof: (related to "Continuous Functions Mapping Compact Domains to Real Numbers Take Maximum and Minimum Values on these Domains") 

2.Corollary: Continuous Functions Mapping Compact Domains to Real Numbers are Bounded