The Distance Between the Values of Two Continuous Maps Is Continuous

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $(X,d_{X})$ and $(Y,d_{Y})$ be metric spaces, let $A\subseteq X$, and let $x,y:A\to Y$ be maps continuous on $A$. Let $h:A\to\mathbb{R}$ be the map with $h(t)=d_{Y}(x(t),y(t))$ for every $t\in A$, which exists and is unique by Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii), and let $\mathbb{R}$ carry the distance $d_{\mathbb{R}}$ of Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces (iv).

(i) $h$ is continuous on $A$.