Functions preserving spheres in \(\mathbb Q_p\) (Q2866260)

The purpose of this paper is to study the functions \(f:A\to\mathbb Q_p\), where \(A\) is an open subset of \(\mathbb Q_p\) and \(0\in A\), which preserve spheres with center at \(0\). Here \(\mathbb Q_p\) denotes the field of \(p\)-adic numbers. Every isometry \(f\) such that \(f(0)=0\) belongs to the set of these functions. The authors show that there exist functions preserving all the spheres with center at \(0\) which are neither continuous nor one-to-one.NEWLINENEWLINE The first main result characterizes the preserving spheres functions which are isometries, proving again a result of \textit{E. Bishop} [J. Ramanujan Math. Soc. 8, No. 1--2, 1--5 (1993; Zbl 0791.11066)]. Next, they give a representation of continuous functions preserving a ball by means of Mahler series. The last section deals with analytic functions preserving spheres.