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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 \(p \not\in S\).
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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