The $p$-adic completion $\mathbb{ Q}_p$ of the rational numbers induces a different absolute value $|\cdot|_p$ than the typical $| \cdot |$ we have on the real numbers. In this paper we compare and contrast functions $f \colon \mathbb{ R}^{ +} \to \mathbb{ R}^{ +}$, for which the composition with the $p$-adic metric $d_p$ generated by $|\cdot|_p$ is still a metric on $\mathbb{ Q}_p$, with the usual metric preserving functions and the functions that preserve the Euclidean metric on $\mathbb{ R}$. In particular, it is shown that $f \circ d_p$ is still an ultrametric on $\mathbb{ Q}_p$ if and only if there is a function $g$ such that $f \circ d_p = g \circ d_p$ and $g \circ d$ is still an ultrametric for every ultrametric $d$. Some general variants of the last statement are also proved.
Categories: math.MG math.NT