Continuous Mapping (Metric Space)/Examples/Composition of Arbitrary Mappings

Examples of Continuous Mappings in the Context of Metric Spaces
Let the following mappings be defined:

where $\R$ and $\R^2$ denote the real number line and real number plane respectively, under the usual (Euclidean) metric.

Then:
 * each of $g, h, k, m$ are continuous
 * $x y = \map {\paren {m \circ k \circ h \circ g} } {x, y}$

where $\circ$ denotes composition of mappings.

Proof
By Composition of Mappings is Associative and General Associativity Theorem, $m \circ j \circ h \circ g$ is well-defined and unambiguous.

Continuity is obvious, but tedious to prove.

Consider $\tuple {x, y} \in \R^2$.

We have: