Metric products and continuation of isotone functions

Let ${\mathbb{R}}_{+}=[0,\mathrm{\infty })$ and let $A\subseteq {\mathbb{R}}_{+}^n$. We have found the necessary and sufficient conditions under which a function $\mathrm{\Phi }:A\to {\mathbb{R}}_{+}$ has an isotone subadditive continuation on ${\mathbb{R}}_{+}^n$. It allows us to describe the metrics, defined on the Cartesian product $X_1\times \mathrm{.}\mathrm{.}\mathrm{.}\times X_n$ of given metric spaces $(X_1,d_{X_1}),\mathrm{.}\mathrm{.}\mathrm{.},(X_n,\mathrm{.}\mathrm{.}\mathrm{.},d_{X_n})$, generated by the isotone metric preserving functions on ${\mathbb{R}}_{+}^n$. It also shows that the isotone metric preserving functions $\mathrm{\Phi }:{\mathbb{R}}_{+}^n\to {\mathbb{R}}_{+}$ coincide with the first moduli of continuity of the nonconstant bornologous functions $g:{\mathbb{R}}_{+}^n\to {\mathbb{R}}_{+}$. We discuss some algebraic properties of sets $X\subseteq \mathbb{R}$ providing the existence of isometric embeddings $f:B\to X$foreverythree-point$B\subseteq $\mathbb{R}$$. In particular, we prove that every finite subset of $\mathbb{R}$ is isometric to some subset of transcendental real numbers.