A function \(f:\mathbb{R}^+\to\mathbb{R}^+\) is said to preserve a metric \(d\) on a set \(X\) if \(f\circ d\) is also a metric on \(X\). The present paper deals with variations on this, by demanding that if \(d\) belongs to a class \(U\) of metrics then \(f\circ d\) belongs to a class \(V\) of metrics. The classes under consideration are those of all metrics, all ultrametrics, and the \(p\)-adic metrics on \(\mathbb{Q}\), for various primes \(p\).
After a review of \(p\)-adic metrics and older results on the preservation of metrics, the final two sections of the paper deal with, in turn, the preservation of \(p\)-adic metrics and of classes of ultrametrics. The former gives conditions on individual functions under which they preserve \(p\)-adic metrics. The latter investigates relations between classes of ultrametrics.