Continuity on a Subset Is Continuity on the Metric Subspace

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, which extends Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces, let $(X,d)$ and $(Y,e)$ be metric spaces and let $A\subseteq X$. Let $d_{A}=d|_{A\times A}$ be the restriction of $d$ to $A\times A$; it is a metric on $A$ (Subsets of a Metric Space: the Restricted Metric, Sequences, Limits in Closed Subsets and Completeness of Closed Subsets (i)), and $d_{A}(a,b)=d(a,b)$ for all $a,b\in A$. Let $f:A\to Y$ be a function.

(i) Continuity at a point. For every $x\in A$: $f$ is continuous at $x$ relative to $A$, as a function from a subset of $(X,d)$ to $(Y,e)$, if and only if $f$ is continuous at $x$ as a function from the metric space $(A,d_{A})$ to $(Y,e)$.

(ii) Continuity on $A$. $f$ is continuous on $A$, as a function from a subset of $(X,d)$ to $(Y,e)$, if and only if $f$ is continuous as a function from the metric space $(A,d_{A})$ to $(Y,e)$.