Proof of Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations

Since $X$ is a set, so is its subclass $S$ (Subclasses of Sets Are Sets, the Union and Power Set of a Set Exist Uniquely, Binary Unions of Sets Are Sets, and the Universal Class Is Proper (i)); so the component and entry maps of Component Maps and Entry Maps (i) and Component Maps and Entry Maps (ii) are defined for maps on $S$. For real numbers, $a\le b<c$ gives $a<c$ (Uniqueness of Least and Greatest Elements, Properties of the Strict Order, and Trichotomy for Total Orders (v), Uniqueness of Least and Greatest Elements, Properties of the Strict Order, and Trichotomy for Total Orders (iv)); such chains are used below without further mention.

Distances. $(\mathbb{R},d_{\mathbb{R}})$ is a metric space by The Absolute Value Metric on the Real Numbers (i). By The Distance on the Real Numbers Is a Map with Values the Absolute Differences (i), The Euclidean Distance on $\mathbb{R}^n$ (i) and The Distance between Real Matrices (i), $d_{\mathbb{R}}(a,b)=|a-b|$, $d_{E}(y,z)=\lVert y-z\rVert$ and $d_{p\times q}(M,N)=\lVert M-N\rVert$. Hence, by Continuous Map Between Metric Spaces (i), a map $F$ from $S$ into $\mathbb{R}$, $\mathbb{R}^{q}$ or $\mathcal{M}_{p\times q}(\mathbb{R})$ is continuous at $s_{0}$ if and only if

(C) for every real $\varepsilon>0$ there is a real $\delta>0$ such that every $u\in S$ with $d(s_{0},u)<\delta$ satisfies $\lVert F(u)-F(s_{0})\rVert<\varepsilon$,

where $\lVert\cdot\rVert$ is read as $|\cdot|$ when $F$ is real-valued. Absolute values and norms are nonnegative, by Rules of Arithmetic and Order in an Ordered Field (x), Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (i) and Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products (i), and $\lVert 0\rVert=0$, by Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (iii) and Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products (iv).

Step 1 (a sum of finitely many deviations). Let $J$ be a finite set, and let $(g_{j})_{j\in J}$ be a family of maps $S\to\mathbb{R}$, each continuous at $s_{0}$ relative to $S$. For $j\in J$ let $a_{j}:S\to\mathbb{R}$ be the map $u\mapsto|g_{j}(u)-g_{j}(s_{0})|=d_{\mathbb{R}}(g_{j}(u),g_{j}(s_{0}))$, and let $j\mapsto a_{j}$ be the family of these maps, a map from $J$ to $\mathbb{R}^{S}$ (Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iv), applied to the map $J\times S\to\mathbb{R}$, $(j,u)\mapsto d_{\mathbb{R}}(g_{j}(u),g_{j}(s_{0}))$, of Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (v) and The Distance on the Real Numbers Is a Map with Values the Absolute Differences (i)). Then $a_{j}(s_{0})=0$, and $a_{j}$ is continuous at $s_{0}$: for $\varepsilon>0$, the $\delta$ of (C) for $g_{j}$ and $\varepsilon$ gives $|a_{j}(u)-a_{j}(s_{0})|=a_{j}(u)<\varepsilon$ for $u\in S$ with $d(s_{0},u)<\delta$. By Finite Sums of Real-Valued Maps: Continuity, Integral and Derivative (i), applied to the family $(a_{j})_{j\in J}$, the map $h=\sum_{j\in J}a_{j}:S\to\mathbb{R}$, $h(u)=\sum_{j\in J}a_{j}(u)$, is continuous at $s_{0}$, and $h(s_{0})=\sum_{j\in J}0=0$ by Sums over Finite Sets in a Commutative Ring and in an Ordered Field: Distributivity, Products of Sums, Vanishing Terms, Sums over Pairs, Expanding Products of Sums, Counting, Comparison, Monotonicity and the Triangle Inequality (v). Hence, by (C) for $h$: for every real $\varepsilon>0$ there is a real $\delta>0$ such that $h(u)\le|h(u)-h(s_{0})|<\varepsilon$ for every $u\in S$ with $d(s_{0},u)<\delta$. The set $[q]$ is finite by Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice (vii), and $[p]\times[q]$ is finite by Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice (v); so Step 1 applies to $J=[q]$ and to $J=[p]\times[q]$.

Proof of (i). Suppose $x$ is continuous at $s_{0}$, and let $k\in[q]$ and $\varepsilon>0$. Take $\delta$ from (C) for $x$ and $\varepsilon$. For $u\in S$ with $d(s_{0},u)<\delta$, by Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (iv) and Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (iv),

$$|x_{k}(u)-x_{k}(s_{0})|=\bigl|\bigl(x(u)-x(s_{0})\bigr)_{k}\bigr|\le\lVert x(u)-x(s_{0})\rVert<\varepsilon ,$$

so $x_{k}$ is continuous at $s_{0}$ by (C). Conversely, suppose every $x_{k}$ is continuous at $s_{0}$, and let $\varepsilon>0$. Take $h$ and $\delta$ from Step 1 for $J=[q]$, the family $k\mapsto x_{k}$ of Component Maps of a Map into $\mathbb{R}^{q}$, Entry Maps of a Map into Matrices, and Maps Assembled from Them (i) and $\varepsilon$. For $u\in S$ with $d(s_{0},u)<\delta$, the components of $x(u)-x(s_{0})$ are $x_{k}(u)-x_{k}(s_{0})$ (Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (iv)), so by Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (iv) and Natural Numbers as Indices: ℕ ⊆ ℕ₀, 1 ∈ ℕ₀, and the Two Readings of an Operation from 1 to n (v),

$$\lVert x(u)-x(s_{0})\rVert\le\sum_{k=1}^{q}|x_{k}(u)-x_{k}(s_{0})|=\sum_{k\in[q]}|x_{k}(u)-x_{k}(s_{0})|=h(u)<\varepsilon .$$

By (C), $x$ is continuous at $s_{0}$.

Proof of (ii). Suppose $A$ is continuous at $s_{0}$, let $i\in[p]$, $j\in[q]$ and $\varepsilon>0$, and take $\delta$ from (C) for $A$ and $\varepsilon$. For $u\in S$ with $d(s_{0},u)<\delta$, $|A_{ij}(u)-A_{ij}(s_{0})|\le d_{p\times q}(A(u),A(s_{0}))<\varepsilon$ by The Real Matrices with the Operator-Norm Distance Form a Complete Metric Space with Entrywise Convergence (ii); so $A_{ij}$ is continuous at $s_{0}$. Conversely, suppose every $A_{ij}$ is continuous at $s_{0}$, and let $\varepsilon>0$. Take $h$ and $\delta$ from Step 1 for $J=[p]\times[q]$, the family $(i,j)\mapsto A_{ij}$ of Component Maps of a Map into $\mathbb{R}^{q}$, Entry Maps of a Map into Matrices, and Maps Assembled from Them (ii) and $\varepsilon$. For $u\in S$ with $d(s_{0},u)<\delta$, the entries of $A(u)-A(s_{0})$ are $A_{kl}(u)-A_{kl}(s_{0})$ (Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (iv)), so by The Operator Norm of Entries, Transposes, Multiples of the Identity, Orthogonal Factors and Diagonal Matrices (iv), Natural Numbers as Indices: ℕ ⊆ ℕ₀, 1 ∈ ℕ₀, and the Two Readings of an Operation from 1 to n (v) and Iterated Operations over Finite Sets: Singletons, Disjoint Unions, Reindexing, Products of Sets, Termwise Combination, Homomorphisms and Intervals (iv),

$$\lVert A(u)-A(s_{0})\rVert\le\sum_{k=1}^{p}\sum_{l=1}^{q}|A_{kl}(u)-A_{kl}(s_{0})|=\sum_{(k,l)\in[p]\times[q]}|A_{kl}(u)-A_{kl}(s_{0})|=h(u)<\varepsilon .$$

By (C), $A$ is continuous at $s_{0}$.

Proof of (iii). Let $\varepsilon>0$ and take $\delta$ from (C) for $x$, respectively $A$, and $\varepsilon$. For $u\in S$ with $d(s_{0},u)<\delta$, $\bigl|\lVert x(u)\rVert-\lVert x(s_{0})\rVert\bigr|\le\lVert x(u)-x(s_{0})\rVert<\varepsilon$ by Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (vii), and $\bigl|\lVert A(u)\rVert-\lVert A(s_{0})\rVert\bigr|\le d_{p\times q}(A(u),A(s_{0}))<\varepsilon$ by The Real Matrices with the Operator-Norm Distance Form a Complete Metric Space with Entrywise Convergence (iii). By (C), both norm maps are continuous at $s_{0}$.

Proof of (iv). Constants. If $F$ is constant, then $\lVert F(u)-F(s_{0})\rVert=\lVert0\rVert=0<\varepsilon$ for every $u\in S$ and every $\varepsilon>0$, so (C) holds with any $\delta>0$.

For each of the other conclusions we assume only that the maps it involves are continuous at $s_{0}$.

Sums, scalar multiples and transposes. For $u\in S$, $k\in[q]$, $i\in[p]$ and $j\in[q]$, by Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (iv), Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (iv) and Transpose of a Real Matrix (i),

$$\bigl(x(u)+y(u)\bigr)_{k}=x_{k}(u)+y_{k}(u),\quad\bigl(\varphi(u)x(u)\bigr)_{k}=\varphi(u)\,x_{k}(u),\quad\bigl(A(u)+B(u)\bigr)_{ij}=A_{ij}(u)+B_{ij}(u),$$ $$\bigl(\varphi(u)A(u)\bigr)_{ij}=\varphi(u)\,A_{ij}(u),\qquad\bigl(A(u)^{\top}\bigr)_{ji}=A_{ij}(u).$$

Whichever of $x$, $y$, $A$ and $B$ are continuous at $s_{0}$ have component, respectively entry, maps continuous at $s_{0}$, by (i) and (ii). For each $k\in[q]$ the component map $(x+y)_{k}$ of $u\mapsto x(u)+y(u)$ is the map $u\mapsto x_{k}(u)+y_{k}(u)$, as both maps have the same values (by the uniqueness in Component Maps and Entry Maps (i)), and likewise for the other component and entry maps displayed above. So the component maps of $u\mapsto x(u)+y(u)$ and $u\mapsto\varphi(u)x(u)$ and the entry maps of $u\mapsto A(u)+B(u)$ and $u\mapsto\varphi(u)A(u)$ are continuous at $s_{0}$ by Continuity of Sums and Products of Real-Valued Functions on a Metric Space (ii) and Continuity of Sums and Products of Real-Valued Functions on a Metric Space (iii), and the entry map of $u\mapsto A(u)^{\top}$ at $(j,i)$ is $A_{ij}$. By (i) and (ii) these five maps are continuous at $s_{0}$.

Products. Let $(F,G,P)$ be one of the triples $(x,y,u\mapsto x(u)\cdot y(u))$, $(A,C,u\mapsto A(u)C(u))$ and $(A,x,u\mapsto A(u)x(u))$, write $F(u)G(u)$ for $P(u)$, put $F_{0}=F(s_{0})$ and $G_{0}=G(s_{0})$, and let $a,b:S\to\mathbb{R}$ be the maps $u\mapsto\lVert F(u)-F_{0}\rVert$ and $u\mapsto\lVert G(u)-G_{0}\rVert$, the distances of $F(u)$ from $F_{0}$ and of $G(u)$ from $G_{0}$ (Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (i) and Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii), with the distance maps of The Summation Map on Real-Valued Maps over a Finite Set, and the Dot Product, Euclidean Norm and Euclidean Distance as Maps (iv) and The Operator-Norm Distance between Real Matrices Is a Map (i)). For $u\in S$,

$$P(u)-P(s_{0})=\bigl(F(u)-F_{0}\bigr)G(u)+F_{0}\bigl(G(u)-G_{0}\bigr),$$

by The Dot Product on $\mathbb{R}^n$ Is Symmetric, Bilinear and Positive Definite (ii) and The Dot Product on $\mathbb{R}^n$ Is Symmetric, Bilinear and Positive Definite (iii) for the dot product, by Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (ii) and Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (iii) for the matrix product and by Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (v) for the matrix-vector product, a difference $M-N$ of matrices being $M+(-1)N$ by Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (iv) and Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (iii). The triangle inequality (Rules of Arithmetic and Order in an Ordered Field (xi) for the real absolute value, Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (vi), Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products (vi)) and the product bounds $|y\cdot z|\le\lVert y\rVert\lVert z\rVert$, $\lVert MN\rVert\le\lVert M\rVert\lVert N\rVert$ and $\lVert Mz\rVert\le\lVert M\rVert\lVert z\rVert$ (The Cauchy-Schwarz Inequality on $\mathbb{R}^n$ (i), Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products (vii), Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products (ii)) give $\lVert P(u)-P(s_{0})\rVert\le a(u)\lVert G(u)\rVert+\lVert F_{0}\rVert\,b(u)$, and $\lVert G(u)\rVert\le b(u)+\lVert G_{0}\rVert$ by the triangle inequality, as $G(u)=(G(u)-G_{0})+G_{0}$. As $a(u)\ge0$,

$$\lVert P(u)-P(s_{0})\rVert\le h(u),\qquad h(u)=a(u)\bigl(b(u)+\lVert G_{0}\rVert\bigr)+\lVert F_{0}\rVert\,b(u)\qquad(u\in S).$$

The map $a$ is continuous at $s_{0}$ with $a(s_{0})=\lVert0\rVert=0$: for $\varepsilon>0$, the $\delta$ of (C) for $F$ and $\varepsilon$ gives $|a(u)-a(s_{0})|=a(u)<\varepsilon$ for $u\in S$ with $d(s_{0},u)<\delta$. Likewise $b$ is continuous at $s_{0}$ with $b(s_{0})=0$. The map $h=a\bigl(b+\lVert G_{0}\rVert\bigr)+\lVert F_{0}\rVert\,b$ is formed with the pointwise operations and constant maps of Pointwise Operations on Real-Valued Maps (i) and Pointwise Operations on Real-Valued Maps (ii); by Continuity of Sums and Products of Real-Valued Functions on a Metric Space (i), Continuity of Sums and Products of Real-Valued Functions on a Metric Space (ii), Continuity of Sums and Products of Real-Valued Functions on a Metric Space (iii) and Continuity of Sums and Products of Real-Valued Functions on a Metric Space (iv), $h$ is continuous at $s_{0}$, and $h(s_{0})=0$. Let $\varepsilon>0$ and take $\delta>0$ such that $|h(u)-h(s_{0})|<\varepsilon$ for $u\in S$ with $d(s_{0},u)<\delta$; for these $u$, $\lVert P(u)-P(s_{0})\rVert\le h(u)=|h(u)-h(s_{0})|<\varepsilon$, as $h(u)\ge0$. By (C), $P$ is continuous at $s_{0}$.

Proof of (v). By Continuous Map Between Metric Spaces (ii), a map is continuous on $S$ if and only if it is continuous at every point of $S$ relative to $S$. Applying (i), (ii), (iii) and (iv) with $s_{0}=s$ for every $s\in S$ gives the statements on $S$.