We obtain characterizations of non-negative functions on $$[0,+\infty )$$
[
0
,
+
∞
)
which preserve some classes of semimetrics. In particular, one of our main results says that for a non-decreasing function $$f:[0,+\infty )\rightarrow [0,+\infty )$$
f
:
[
0
,
+
∞
)
→
[
0
,
+
∞
)
the following statements are equivalent: (i) for any semimetric space (X, d), if d satisfies the relaxed polygonal inequality, then so does $$f\circ d$$
f
∘
d
; (ii) there exist a constant $$c\geqslant 1$$
c
⩾
1
and a subadditive function $$g:[0,+\infty ) \rightarrow [0,+\infty )$$
g
:
[
0
,
+
∞
)
→
[
0
,
+
∞
)
such that $$g^{-1}\left( \{ 0 \} \right) = \{ 0 \}$$
g
-
1
{
0
}
=
{
0
}
and $$g\leqslant f \leqslant cg$$
g
⩽
f
⩽
c
g
. We also obtain a complete characterization of functions preserving regularity of a semimetric space in the sense of Bessenyei and Páles. Finally, we give another proof of the theorem of Pongsriiam and Termwuttipong on functions transforming metrics into ultrametrics.
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