The Real Line: Standing Notation and Background for Calculus on Intervals

(i) This setting extends Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices: its conventions and notation are in force.

(ii) $\mathbb{R}$, its order, its operations, its elements $0$ and $1$ and the notation for them are those of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (ii).

(iii) Bounds, suprema and infima of subsets of $\mathbb{R}$, and their existence, are those of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (vi), The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (vii) and The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (viii). Positive slack and the approximation of suprema and infima by elements of $S$ are as in Arbitrary Positive Slack, and Approximation of Suprema and Infima, in the Real Numbers.

(iv) A subset $E\subseteq\mathbb{R}$ carries the restriction $d_{E}=d_{\mathbb{R}}|_{E\times E}$, a metric on $E$ by Subsets of a Metric Space: the Restricted Metric, Sequences, Limits in Closed Subsets and Completeness of Closed Subsets (i), with $d_{E}(s,t)=|s-t|$ for $s,t\in E$. A map $f:E\to\mathbb{R}$ is read as a map from the subset $E$ of $(\mathbb{R},d_{\mathbb{R}})$ into $(\mathbb{R},d_{\mathbb{R}})$, or as a map from $(E,d_{E})$ into $(\mathbb{R},d_{\mathbb{R}})$; for continuity and uniform continuity the two readings agree, by (vi). A subset of $\mathbb{R}$ is compact if it is compact in $(\mathbb{R},\mathcal{T}_{d_{\mathbb{R}}})$.

(v) Intervals, the bounded intervals $[a,b]$, $(a,b)$, $[a,b)$, $(a,b]$ and the rays $(a,\infty)$, $[a,\infty)$, $(-\infty,b)$, $(-\infty,b]$, for $a,b\in\mathbb{R}$, are as in Intervals of the Real Line, and they obey Basic Facts about Intervals of the Real Line.

(vi) Let $E\subseteq\mathbb{R}$, let $f:E\to\mathbb{R}$ and let $x\in E$. $f$ is continuous at $x$ and $f$ is continuous on $E$ mean that $f$ is continuous at $x$ relative to $E$, respectively continuous on $E$, for $(X,d_{X})=(Y,d_{Y})=(\mathbb{R},d_{\mathbb{R}})$ and $A=E$. By Continuity of Real Functions on Subsets of ℝ in ε-δ Form (i), $f$ is continuous at $x$ if and only if for every real $\varepsilon>0$ there is a real $\delta>0$ such that every $y\in E$ with $|y-x|<\delta$ satisfies $|f(y)-f(x)|<\varepsilon$; and by Continuity of Real Functions on Subsets of ℝ in ε-δ Form (ii), $f$ is continuous on $E$ if and only if this holds at every point of $E$ in place of $x$. Likewise, by Uniformly Continuous Map Between Metric Spaces (i), $f$ is uniformly continuous on $E$ if and only if for every real $\varepsilon>0$ there is a real $\delta>0$ such that all $y,z\in E$ with $|y-z|<\delta$ satisfy $|f(y)-f(z)|<\varepsilon$. Since $d_{E}(s,t)=d_{\mathbb{R}}(s,t)$ for $s,t\in E$ by (iv), these conditions are unchanged when $f$ is read as a map from $(E,d_{E})$, with $A=X=E$. For $F\subseteq E$ with $x\in F$, if $f$ is continuous at $x$, respectively continuous on $E$, then its restriction $f|_{F}:F\to\mathbb{R}$ is continuous at $x$, respectively continuous on $F$, by Restricting a Continuous Map to a Smaller Subset Preserves Continuity (i) and Restricting a Continuous Map to a Smaller Subset Preserves Continuity (ii). Identity maps of subsets of a metric space are continuous on them, by Continuity of Identity Maps, Compositions and Reciprocals (i). Let $(X,d_{X})$, $(Y,d_{Y})$ and $(Z,d_{Z})$ be metric spaces, let $A\subseteq X$ and $B\subseteq Y$, let $\varphi:A\to Y$ satisfy $\varphi(u)\in B$ for every $u\in A$, and let $g:B\to Z$; then, by Continuity of Identity Maps, Compositions and Reciprocals (ii), the composition $g\circ\varphi:A\to Z$ is continuous at $v\in A$ relative to $A$ if $\varphi$ is continuous at $v$ relative to $A$ and $g$ is continuous at $\varphi(v)$ relative to $B$, and $g\circ\varphi$ is continuous on $A$ if $\varphi$ is continuous on $A$ and $g$ is continuous on $B$. The reciprocal $1/f$ of a map $f:E\to\mathbb{R}$ without zeros is continuous at $x$, respectively on $E$, when $f$ is, by Continuity of Identity Maps, Compositions and Reciprocals (iii).

(vii) $\mathbb{N}\subseteq\mathbb{N}_{0}\subseteq\mathbb{R}$ and the Archimedean property are those of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (v) and The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (ix).

(viii) Sequences in $\mathbb{R}$, their convergence, limits and the Cauchy property are those of Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces (iv).

(ix) For $s\in\mathbb{R}$ with $s\ge0$, $\sqrt{s}$ is the square root of $s$: the $r\in\mathbb{R}$ with $r\ge0$ and $r^{2}=s$, which exists and is unique by Existence and Uniqueness of Nonnegative k-th Roots of Nonnegative Real Numbers (i). For $s,t\in\mathbb{R}$ with $s\ge0$ and $t\ge0$, $s\le t$ if and only if $\sqrt{s}\le\sqrt{t}$, by Properties of k-th Roots: Inverse to Powers, Absolute Values, Products, Monotonicity, Subadditivity, Differences and Limits (iv).

(x) For $a,b\in\mathbb{R}$ with $a\le b$, $[a,b]$ is compact, by Closed Interval $[a,b]$ is Compact in $\mathbb{R}$ (i). A continuous map on a compact subset of a metric space, or on a compact metric space, is uniformly continuous, by Heine-Cantor Theorem: Continuity on a Compact Metric Space or a Compact Subset Implies Uniform Continuity (ii) and Heine-Cantor Theorem: Continuity on a Compact Metric Space or a Compact Subset Implies Uniform Continuity (i). A continuous map $f:[a,b]\to\mathbb{R}$ with $a\le b$ is bounded and attains a least and a greatest value on $[a,b]$, by Extreme Value Theorem on a Closed Real Interval (i) and Extreme Value Theorem on a Closed Real Interval (ii). A continuous map $f:[a,b]\to\mathbb{R}$ with $a<b$ takes every value between $f(a)$ and $f(b)$, by Intermediate Value Theorem on a Closed Real Interval, and Continuous Images of Intervals (i), and the image of an interval under a continuous map into $\mathbb{R}$ is an interval, by Intermediate Value Theorem on a Closed Real Interval, and Continuous Images of Intervals (ii).