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Arithmetic subderivatives: $$p$$-adic discontinuity and continuity. (English) Zbl 1453.11004
Summary: In a previous paper, we proved that the arithmetic subderivative $$D_S$$ is discontinuous at any rational point with respect to the ordinary absolute value. In the present paper, we study this question with respect to the $$p$$-adic absolute value. In particular, we show that $$D_S$$ is in this sense continuous at the origin if $$S$$ is finite or $$p \not\in S$$.
##### MSC:
 11A25 Arithmetic functions; related numbers; inversion formulas 11S82 Non-Archimedean dynamical systems 26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
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