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Maximum-minimum theorem

Theorem

(Maximum-Minimum Theorem). Let $(X,d)$ be a metric space and let $f:A\to ℝ$ be a continuous function with domain $A\subset X$. For any compact set $B\subset A$, $f$ is bounded on $B$ and attains its supremum and infimum on $B$. That is, $\mid f(x)\mid \le M$ for some real number $M<\mathrm{\infty }$ and there exist points $x_0$ and $x_1$ in $B$ such that $f(x_0)={\inf }_{x\in B}f(x)$ and $f(x_1)={\sup }_{x\in B}f(x)$.