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Proof

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

• Let $X$ be a metric spaces.
• Let $D\subset X$ be a compact subset.
• Let $f:D\mapsto \mathbb R$ be a continuous function mapping the domain $D$ to the real numbers $\mathbb R$.
• We have to show that there are points \(p,q\in X\), for which the function $f$ takes the maximum and the maximum values of the image $f(D)$, i.e. $f(p)=\max(f(D))$ and $f(q)=\min(f(D)).$
  • Since $f$ is continuous and \(D\subset X\) is compact by hypothesis, and since real numbers form a metric space, the image \(f( D)\subset \mathbb R\), is also compact, according to the corresponding proposition.
  • Because compact subsets of real numbers contain their maxima and minima, it follows that $\max(f(D))\in f(D)$ and $\min(f(D))\in f(D)$.
  • Therefore, the fibers $P:=f^{-1}(\max(f(D)))$ and $Q:=f^{-1}(\min(f(D)))$ are not-empty subsets $P\subseteq X$ and $Q\subseteq X$.
  • It follows that \(p,q\in X\) exist.
    ∎

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References

Bibliography

1. Forster Otto: "Analysis 2, Differentialrechnung im \(\mathbb R^n\), Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984