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

In the setting of Euclidean Space and Real Matrices: Standing Notation, let $(X,d)$ be a metric space, let $S\subseteq X$ and let $p,q,r\in\mathbb{N}$. $\mathbb{R}$ carries the metric $d_{\mathbb{R}}$, $\mathbb{R}^{q}$ the metric $d_{E}$ and $\mathcal{M}_{p\times q}(\mathbb{R})$ the metric $d_{p\times q}$, as in Euclidean Space and Real Matrices: Standing Notation (ii) and Euclidean Space and Real Matrices: Standing Notation (vi). For $s_{0}\in S$, a map from $S$ into one of these spaces is continuous at $s_{0}$ if it is continuous at $s_{0}$ relative to $S$; it is continuous on $S$ as in Continuous Map Between Metric Spaces (ii). For $x:S\to\mathbb{R}^{q}$ and $k\in[q]$, $x_{k}:S\to\mathbb{R}$ is the component map $u\mapsto x(u)_{k}$ of Component Maps and Entry Maps (i); for $A:S\to\mathcal{M}_{p\times q}(\mathbb{R})$, $i\in[p]$ and $j\in[q]$, $A_{ij}:S\to\mathbb{R}$ is the entry map $u\mapsto A(u)_{ij}$ of Component Maps and Entry Maps (ii). In (i), (ii), (iii) and (iv), let $s_{0}\in S$.

(i) A map $x:S\to\mathbb{R}^{q}$ is continuous at $s_{0}$ if and only if $x_{k}$ is continuous at $s_{0}$ for every $k\in[q]$.

(ii) A map $A:S\to\mathcal{M}_{p\times q}(\mathbb{R})$ is continuous at $s_{0}$ if and only if $A_{ij}$ is continuous at $s_{0}$ for all $i\in[p]$ and $j\in[q]$.

(iii) If $x:S\to\mathbb{R}^{q}$ is continuous at $s_{0}$, then so is $u\mapsto\lVert x(u)\rVert$; if $A:S\to\mathcal{M}_{p\times q}(\mathbb{R})$ is continuous at $s_{0}$, then so is $u\mapsto\lVert A(u)\rVert$.

(iv) Every constant map from $S$ into $\mathbb{R}^{q}$ or into $\mathcal{M}_{p\times q}(\mathbb{R})$ is continuous at $s_{0}$. Let $x,y:S\to\mathbb{R}^{q}$, $\varphi:S\to\mathbb{R}$, $A,B:S\to\mathcal{M}_{p\times q}(\mathbb{R})$ and $C:S\to\mathcal{M}_{q\times r}(\mathbb{R})$. If $x$ and $y$ are continuous at $s_{0}$, then so are $u\mapsto x(u)+y(u)$ and $u\mapsto x(u)\cdot y(u)$; if $\varphi$ and $x$ are, then so is $u\mapsto\varphi(u)\,x(u)$; if $A$ and $B$ are, then so is $u\mapsto A(u)+B(u)$; if $\varphi$ and $A$ are, then so is $u\mapsto\varphi(u)\,A(u)$; if $A$ and $C$ are, then so is $u\mapsto A(u)\,C(u)$, into $\mathcal{M}_{p\times r}(\mathbb{R})$; if $A$ and $x$ are, then so is $u\mapsto A(u)\,x(u)$, into $\mathbb{R}^{p}$; and if $A$ is, then so is $u\mapsto A(u)^{\top}$, into $\mathcal{M}_{q\times p}(\mathbb{R})$.

(v) Each of (i), (ii), (iii) and (iv) holds with continuous at $s_{0}$'' replaced by continuous on $S$'' throughout.