A Countable Uniformly Dense Family of Lipschitz Functions on a Compact Metric Space

Let $(K,d)$ be a metric space with $K$ nonempty, and let $\mathcal{T}_d$ be the collection of subsets of $K$ that are open in $(K,d)$, which is a topology on $K$ by Metric Open Sets Form a Topology. Assume that $K$ is compact in $(K,\mathcal{T}_d)$.

Let $\mathbb{R}$ denote the real numbers, with the addition, multiplication, identities, additive inverses, multiplicative inverses and order of their ordered field structure; for $s,t\in\mathbb{R}$ write $s-t$ for $s+(-t)$, write $s<t$ to mean that $s\le t$ and $s\ne t$, write $s^{-1}$ for the multiplicative inverse of $s$ when $s\ne 0$, let $|s|$ be the absolute value of $s$, and let $d_{\mathbb{R}}$ be given by $d_{\mathbb{R}}(s,t)=|s-t|$, which is a metric on $\mathbb{R}$ by The Absolute Value Metric on the Real Line.

Then the following hold.

1. (Uniform approximation by Lipschitz functions) For every map $f:K\to\mathbb{R}$ that is continuous on $K$ as a map from $(K,d)$ to $(\mathbb{R},d_{\mathbb{R}})$, and for every $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$, there exists a map $g:K\to\mathbb{R}$ that is Lipschitz as a map from $(K,d)$ to $(\mathbb{R},d_{\mathbb{R}})$ and satisfies

$$|f(x)-g(x)|\le\varepsilon\qquad\text{for every }x\in K .$$

2. (A countable uniformly dense family) There exists a countable set $\mathcal{G}$ whose elements are maps from $K$ to $\mathbb{R}$, such that every $g\in\mathcal{G}$ is Lipschitz as a map from $(K,d)$ to $(\mathbb{R},d_{\mathbb{R}})$ and is bounded, and such that for every map $f:K\to\mathbb{R}$ continuous on $K$ and every $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$ there exists $g\in\mathcal{G}$ with

$$|f(x)-g(x)|\le\varepsilon\qquad\text{for every }x\in K .$$