Continuity of Identity Maps, Compositions and Reciprocals

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, metric spaces are as in Metric Space (vi), and continuity at a point relative to a subset and on a subset are as in Continuous Map Between Metric Spaces (i) and Continuous Map Between Metric Spaces (ii).

(i) Let $(X,d)$ be a metric space and $A\subseteq X$. The identity map $A\to X$, $u\mapsto u$ (Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (v)), is continuous on $A$.

(ii) Let $(X,d_{X})$, $(Y,d_{Y})$ and $(Z,d_{Z})$ be metric spaces, let $A\subseteq X$ and $B\subseteq Y$, let $f:A\to Y$ satisfy $f(u)\in B$ for every $u\in A$, and let $g:B\to Z$. Then $f$ is also a map $A\to B$, by Functions, Values of a Function, and Functions from One Class to Another (iii) and Basic Properties of Functions: Equality, Composition, Identity, Inverse and Restriction (v), and $g\circ f:A\to Z$ is the composition, with $(g\circ f)(u)=g(f(u))$ for $u\in A$, by Basic Properties of Functions: Equality, Composition, Identity, Inverse and Restriction (ii). If $f$ is continuous at $x\in A$ relative to $A$ and $g$ is continuous at $f(x)$ relative to $B$, then $g\circ f$ is continuous at $x$ relative to $A$. If $f$ is continuous on $A$ and $g$ is continuous on $B$, then $g\circ f$ is continuous on $A$.

(iii) Let $d_{\mathbb{R}}$ be the distance on $\mathbb{R}$, a metric by The Absolute Value Metric on the Real Numbers (i), and read maps from a subset of $\mathbb{R}$ into $\mathbb{R}$ as maps from that subset of $(\mathbb{R},d_{\mathbb{R}})$ into $(\mathbb{R},d_{\mathbb{R}})$. Let $E\subseteq\mathbb{R}$, let $g:E\to\mathbb{R}$ satisfy $g(t)\neq0$ for every $t\in E$, and let $1/g:E\to\mathbb{R}$ be the map $t\mapsto1/g(t)$ (Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (v)), with quotients as in Negatives, Differences, Reciprocals and Quotients (ii). If $g$ is continuous at $x\in E$ relative to $E$, then so is $1/g$. If $g$ is continuous on $E$, then so is $1/g$.