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

Let $A \subseteq \mathbb{R}$ carry the subspace metric of the usual metric $d_{\mathbb{R}}(x,y) = |x-y|$ of $\mathbb{R}$, that is $d_A(x,y) = |x - y|$ for $x, y \in A$ (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, Isometry, isometric embedding, and the subspace metric on a subset), and let $f : A \to \mathbb{R}$, regarded also as a map of metric spaces $(A, d_A) \to (\mathbb{R}, d_{\mathbb{R}})$. Then the $\mathbb{R}$-native notions of this page and the metric-space notions of the earlier pages are the same notions, in the following five senses.

1. Continuity. For every $c \in A$: $f$ is continuous at $c$ in the sense of 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 if and only if $f$ is continuous at $c$ in the sense of Continuity of a map between metric spaces, at a point and globally, in the $\varepsilon$-$\delta$ form. Consequently $f$ is continuous on $A$ if and only if it is continuous as a map of metric spaces.
2. Uniform continuity. $f$ is uniformly continuous on $A$ in the sense of Uniform continuity of $f : A \to \mathbb{R}$: one $\delta$ serving every pair of points of $A$ if and only if it is uniformly continuous as a map of metric spaces (Uniform continuity of a map of metric spaces: one $\delta$ serving every point).
3. Lipschitz. For a real $L \ge 0$: $f$ is Lipschitz with constant $L$ as a map of metric spaces (Lipschitz map, $\alpha$-Hölder map for rational $0 < \alpha \le 1$, and contraction) if and only if $|f(x) - f(x')| \le L\,|x - x'| \qquad \text{for all } x, x' \in A .$ This displayed condition is what "$f$ is Lipschitz with constant $L$" means for a real function on $A$ in this library; no second definition is made.
4. Hölder. For a rational $\alpha$ with $0 < \alpha \le 1$ and a real $C \ge 0$: $f$ is $\alpha$-Hölder with constant $C$ as a map of metric spaces if and only if $|f(x) - f(x')| \le C\,|x - x'|^{\alpha} \qquad \text{for all } x, x' \in A ,$ the power being the rational power of a nonnegative base (Rational powers $a^r$ of a positive base).
5. Compactness, in both senses used in this library. For $K \subseteq \mathbb{R}$ with the subspace metric $d_K$:
   • $K$ is compact in the open-cover sense of Open cover, subcover, compact subset of $\mathbb{R}$ (every open cover has a finite subcover), and sequentially compact subset — every family of open subsets of $\mathbb{R}$ covering $K$ has a finite subfamily covering $K$ — if and only if the metric space $(K, d_K)$ is compact (Open cover, subcover, compact metric space, and compact subset of a metric space);
   • $K$ is sequentially compact in the sense of Open cover, subcover, compact subset of $\mathbb{R}$ (every open cover has a finite subcover), and sequentially compact subset if and only if $(K, d_K)$ is sequentially compact as a metric space (Countably compact, sequentially compact and limit point compact metric spaces).

Two consequences are recorded, since they are the reason the dictionary is stated as a lemma rather than as a remark.

6. The regularity hierarchy transfers verbatim. By clauses 1 to 4 and Contraction implies Lipschitz implies uniformly continuous implies continuous; every Hölder map is uniformly continuous, and a Lipschitz map on a bounded space is Hölder for every exponent: a Lipschitz $f : A \to \mathbb{R}$ is uniformly continuous on $A$; an $\alpha$-Hölder $f$ with rational $0 < \alpha \le 1$ is uniformly continuous on $A$; a uniformly continuous $f$ is continuous on $A$; and if $A$ is nonempty and bounded, a Lipschitz $f$ is $\alpha$-Hölder for every rational $\alpha$ with $0 < \alpha \le 1$. No strictness is claimed here, and none is claimed there.
7. Cauchy sequences transfer. A sequence $(x_k)$ with terms in $A$ is Cauchy in $(A, d_A)$ (Cauchy sequence in a metric space) if and only if it is Cauchy as a sequence of reals (Limits and Cauchy sequences of reals); so by clause 2 and A uniformly continuous map sends Cauchy sequences to Cauchy sequences, a uniformly continuous $f : A \to \mathbb{R}$ carries Cauchy sequences of $A$ to Cauchy sequences of $\mathbb{R}$.

Why this lemma exists, and why it is a lemma. Three results of this page — The image of a compact subset of $\mathbb{R}$ under a continuous real function is compact, Extreme value theorem: a continuous real function on a nonempty compact subset of $\mathbb{R}$ attains a greatest and a least value and Heine-Cantor in $\mathbb{R}$: a continuous real function on a compact subset of $\mathbb{R}$ is uniformly continuous, proved $\mathbb{R}$-natively from sequential compactness — are stated a second time here, having already been proved metric-generally as The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset, A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value and Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous. The duplication is deliberate: the $\mathbb{R}$-native proofs run through A subset of $\mathbb{R}$ is compact if and only if it is closed and bounded and A subset of $\mathbb{R}$ is compact iff it is sequentially compact, which are order-based, while the metric proofs run through the cover machinery of metric spaces. This item is the single place in the library where that duplication is acknowledged, and clauses 1 and 5 are what make the two families of statements literally the same statements. It is a lemma, and not a remark, precisely so that later pages can cite it and move between the two vocabularies.

Clause 5 closes a second seam. The phrase compact subset of $\mathbb{R}$ is defined twice in this library — metrically, as compactness of the metric subspace (Open cover, subcover, compact metric space, and compact subset of a metric space), and $\mathbb{R}$-natively, by covers by open subsets of $\mathbb{R}$ (Open cover, subcover, compact subset of $\mathbb{R}$ (every open cover has a finite subcover), and sequentially compact subset) — and until this clause no item asserted that the two agree.

• Nothing here is a new theorem, and that is the point. Every clause is an identification of two forms of words, and the only clause with any content beyond substitution is 5, which needs A subset of a metric space is open in the subspace metric exactly when it is the trace of an open set of the ambient space, and it is compact as a metric space in its own right exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it to move between covers by relatively open sets and covers by open subsets of $\mathbb{R}$, and needs 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 to know that the metric topology of $\mathbb{R}$ is the topology of Open subset of $\mathbb{R}$ (every point has a neighbourhood inside it), closed subset (complement open), and clopen. A reader who takes those two identifications for granted is taking for granted exactly what this library refuses to leave unsaid.

• The hierarchy of clause 6 is not strict by fiat, and the witnesses live on the companion page. Contraction implies Lipschitz implies uniformly continuous implies continuous; every Hölder map is uniformly continuous, and a Lipschitz map on a bounded space is Hölder for every exponent asserts the four implications and claims no converse. That none of them reverses for real functions is witnessed here: On $[0,1]$ the function $x^{\beta}$ is $\beta$-Hölder and is $\alpha$-Hölder for no rational $\alpha > \beta$, so the Hölder classes are strictly nested ↗ gives, for rationals $0 < \beta < \alpha \le 1$, a function on $[0,1]$ that is $\beta$-Hölder and not $\alpha$-Hölder, and in particular ($\alpha = 1$) is uniformly continuous and not Lipschitz; and $x \mapsto 1/x$ is continuous on $(0,1)$ and not uniformly continuous there, the pairs $1/(k+2)$ and $1/(k+3)$ defeating every $\delta$ ↗ gives a continuous function that is not uniformly continuous. Those two items are links, not dependencies: they are examples on the companion page, and nothing on this page rests on them.

• What the dictionary does not say. It does not say that the two proofs of a duplicated theorem are the same proof, and they are not: the $\mathbb{R}$-native ones use the order of $\mathbb{R}$ and spend no choice beyond what is named in each item, while the metric ones use covers and, where the equivalence of the compactness variants is invoked, countable or dependent choice. What the dictionary establishes is that the two statements coincide, so that a later page may use whichever proof it prefers and cite whichever form it needs.