A Continuous Function on a Compact Set Admits a Nondecreasing Modulus of Continuity

Let $(M,d)$ be a metric space, equipped with the collection of all subsets that are open in $(M,d)$, which is a topology by The Open Subsets of a Metric Space Form a Topology. Let $\mathbb{R}$ be the ordered field of real numbers, let $|\cdot|$ be the absolute value on $\mathbb{R}$, let $(\mathbb{R},d_{\mathbb{R}})$ be the real line, so that $d_{\mathbb{R}}(s,t)=|s-t|$, and let $T=\{t\in\mathbb{R}:0\le t\}$.

Let $K\subseteq M$ be nonempty and compact in $M$, and let $f:K\to\mathbb{R}$ be continuous on $K$, as a map into $(\mathbb{R},d_{\mathbb{R}})$.

Then there exists a modulus of continuity $\omega:T\to\mathbb{R}$ with the following two properties.

1. (Monotonicity)(i) $\omega(s)\le\omega(t)$ for all $s,t\in T$ with $s\le t$.

2. (Domination)(ii) For all $x,y\in K$ and every $t\in T$ with $d(x,y)\le t$,

$$|f(x)-f(y)|\le\omega(t).$$