The distance $\psi(x) = d(x, \mathbb{Z})$ from a real number to the integers is $1$-Lipschitz, hence uniformly continuous, takes values in $[0,1/2]$, and vanishes exactly on $\mathbb{Z}$

Take $\mathbb{R}$ with its usual metric $d(x,y) = |x-y|$ (The absolute value makes $\mathbb{R}$ a metric space: $d(x,y) = |x-y|$ is a metric, its open balls are the intervals $(x-r, x+r)$, and it is unbounded), identify $\mathbb{Z}$ with its canonical copy inside $\mathbb{R}$ (The integers as equivalence classes of pairs of naturals, The naturals embed in the integers, The integers embed in the rationals, The rationals embed densely in the reals), and let

$\psi(x) \;:=\; d(x, \mathbb{Z}) \;=\; \inf\{\, |x - n| \ : \ n \in \mathbb{Z} \,\}$

be the distance from $x$ to the nonempty set $\mathbb{Z}$ (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space). Then:

1. $\psi$ is $1$-Lipschitz on $\mathbb{R}$: $|\psi(x) - \psi(y)| \le |x-y|$ for all real $x, y$. Consequently $\psi$ is uniformly continuous on $\mathbb{R}$ (Uniform continuity of $f : A \to \mathbb{R}$: one $\delta$ serving every pair of points of $A$) and continuous on $\mathbb{R}$ (Continuity of $f : A \to \mathbb{R}$ at a point of $A$ and on $A$: the $\varepsilon$-$\delta$ condition, its agreement with $\lim_{x \to c} f(x) = f(c)$ at a limit point, and continuity at an isolated point).
2. The infimum is attained, and computed. Writing $m := \lfloor x \rfloor$ for the integer part of $x$ (Integer part: for every real $x$ there is exactly one integer $m$ with $m \le x < m + 1$) and $t := x - m$, so that $0 \le t < 1$, $\psi(x) \;=\; \min\{\, t,\ 1-t \,\} \;=\; \min\bigl\{\, |x-m|,\ |x-(m+1)| \,\bigr\} ,$ so $\psi(x) = |x - n|$ for $n = m$ or $n = m+1$.
3. Range. $0 \le \psi(x) \le 1/2$ for every real $x$ (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length).
4. Zero set. $\psi(x) = 0$ if and only if $x \in \mathbb{Z}$.

Why this example is here. It is the standard uniformly continuous function of this track that is not defined by a formula in the field operations, and it is obtained from the metric machinery rather than rebuilt: claim 1 is $|d(x,A) - d(y,A)| \le d(x,y)$, so the distance to a fixed nonempty set is $1$-Lipschitz applied to $A := \mathbb{Z}$ in the metric space $\mathbb{R}$, transported to a statement about a real function by Dictionary: for $A \subseteq \mathbb{R}$ with the metric $d(x,y) = |x-y|$, continuity and uniform continuity of $f : A \to \mathbb{R}$ agree with the metric-space notions, the Lipschitz and Hölder conditions are the metric ones instantiated, and a subset of $\mathbb{R}$ is compact in the open-cover sense of $\mathbb{R}$ exactly when it is a compact metric subspace, clause 3. Only claims 2 to 4, which compute the value, need an argument of their own.

The same function is computed elsewhere, and nothing here depends on that. The trigonometry-free oscillator $\psi(x) = \inf_{n \in \mathbb{Z}} |x - n|$ is well defined and attained at a nearest integer, takes values in $[0, 1/2]$, vanishes exactly on $\mathbb{Z}$, equals $1/2$ at half-integers, and is $1$-periodic introduces $\psi$ on the companion page of The $\varepsilon$-$\delta$ limit $\lim_{x \to c} f(x) = L$ of $f : A \to \mathbb{R}$ at a limit point $c$ of $A$ and proves the same computation together with $1$-periodicity and the value at half-integers. That item lives on an examples page, which is a leaf of the dependency graph, so no item may rest on it; the verification below is therefore self-contained, and the duplication is deliberate rather than an oversight.

• No completeness of $\mathbb{R}$ is spent on the infimum here. Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space produces $d(x,\mathbb{Z})$ from the greatest-lower-bound property, but step 3.1 does not need that route: it exhibits an element of the set of distances that is also a lower bound, which is the definition of the infimum read directly (Greatest lower bound (infimum)). Completeness does enter once, through Integer part: for every real $x$ there is exactly one integer $m$ with $m \le x < m + 1$, whose existence half is the Archimedean property.

• Why "nearest integer" is a theorem and not a phrase. The words presuppose that a nearest integer exists, and that is exactly what steps 2.2 and 2.3 establish. When $t = 1/2$ there are two nearest integers, $m$ and $m+1$, and the formula $\min\{t,1-t\}$ is indifferent to which is taken, so nothing is selected.

• What this contributes to the hierarchy. $\psi$ is Lipschitz, hence uniformly continuous (Dictionary: for $A \subseteq \mathbb{R}$ with the metric $d(x,y) = |x-y|$, continuity and uniform continuity of $f : A \to \mathbb{R}$ agree with the metric-space notions, the Lipschitz and Hölder conditions are the metric ones instantiated, and a subset of $\mathbb{R}$ is compact in the open-cover sense of $\mathbb{R}$ exactly when it is a compact metric subspace, clause 6), and it is bounded and not monotone; so it is a uniformly continuous function that is neither a polynomial nor eventually constant, and it is the natural domain-wide example to set beside $x \mapsto 1/x$ is continuous on $(0,1)$ and not uniformly continuous there, the pairs $1/(k+2)$ and $1/(k+3)$ defeating every $\delta$, where uniform continuity fails.