Uniform Continuity Along a Compact Subset of the Domain

Let $(X,d)$ be a metric space, equipped with the collection of all subsets open in $(X,d)$, which is a topology by The Open Subsets of a Metric Space Form a Topology. Let $\mathbb{R}$ be the real numbers with the order $\le$ of their ordered field structure, write $s<t$ to mean that $s\le t$ and $s\ne t$, let $|s|$ be the absolute value of $s$, and let $d_{\mathbb{R}}$ be given by $d_{\mathbb{R}}(s,t)=|s-t|$, a metric on $\mathbb{R}$ by The Absolute Value Metric on the Real Numbers.

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

Then for every $\eta\in\mathbb{R}$ with $0<\eta$ there is $\theta\in\mathbb{R}$ with $0<\theta$ such that

$$|f(z)-f(x)|<\eta$$

for every $x\in K$ and every $z\in\Omega$ with $d(x,z)<\theta$.

Note that $z$ is required only to lie in $\Omega$, not in $K$.