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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 compactsubset 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 $$\exists p,q\in X:~f(p)=\max(f(D)),\quad f(q)=\min(f(D)).$$