Restricting a Continuous Map to a Smaller Subset Preserves Continuity

In the setting of Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces, let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, let $A$ and $B$ be subsets of $X$ with $B\subseteq A$, let $f:A\to Y$ be a map, and let $f|_{B}$ be the restriction of $f$ to $B$, which is a map $f|_{B}:B\to Y$ by Domains and Restrictions of Maps (ii).

(i) Let $x\in B$. If $f$ is continuous at $x$ relative to $A$, then $f|_{B}$ is continuous at $x$ relative to $B$.

(ii) If $f$ is continuous on $A$, then $f|_{B}$ is continuous on $B$.