Extreme Value Theorem on a Compact Subset of a Metric Space

Let $(X,d)$ be a metric space, equipped with the collection of all subsets that are open in $(X,d)$, which is a topology by Metric Open Sets Form a Topology. Let $K\subseteq X$ be nonempty and compact in $X$. Let $\mathbb{R}$ be the set of real numbers with the order $\le$ of its ordered field structure, and let $|\cdot|$ be the 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