Extreme Value Theorem on a Compact Subset of a Metric Space

Let $(X,d)$ be a \reftext{def:metric-space-2026a}{metric space}, equipped with the collection of all subsets that are \reftext{def:open-subset-metric-space-2026a}{open in $(X,d)$}, which is a topology by \ref{thm:metric-open-sets-form-topology-2026a}. Let $K\subseteq X$ be nonempty and \reftext{def:compact-space-and-subset-2026a}{compact in $X$}. Let $\mathbb{R}$ be the set of \reftext{def:real-numbers-c54-2026c}{real numbers} with the order $\le$ of its \reftext{def:ordered-field-c54-2026b}{ordered field} structure, and let $|\cdot|$ be the \reftext{def:absolute-value-ordered-field-2026a}{absolute value} on $\mathbb{R}$.

Let $f:K\to\mathbb{R}$ be a function with the following continuity property: for every $x\in K$ and every real number $\varepsilon>0$ there exists a real number $\delta>0$ such that every $y\in K$ with $d(x,y)<\delta$ satisfies

Then there exist $x_{\min},x_{\max}\in K$ such that