P-adic metric preserving functions and their analogues

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.