  We first prove a version of Tietze-Urysohn's theorem for proper functions
taking values in non-negative real numbers defined on $\sigma$-compact locally
compact Hausdorff spaces. As its application, we prove an extension theorem of
proper metrics, which states that if $X$ is a $\sigma$-compact locally compact
space, $A$ is a closed subset of $X$, and $d$ is a proper metric on $A$ that
generates the same topology of $A$, then there exists a proper metric on $X$
such that $D$ generates the same topology of $X$ and $D|_{A^{2}}=d$. Moreover,
if $A$ is a proper retraction, we can choose $D$ so that $(A, d)$ is
quasi-isometric to $(X, D)$. We also show analogues of theorems explained above
for ultrametric spaces.
