  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.
