Journal of Integer Sequences, Vol. 22 (2019), Article 19.7.4

Arithmetic Subderivatives: Discontinuity and Continuity

Pentti Haukkanen and Jorma K. Merikoski
Faculty of Information Technology and Communication Sciences
FI-33014 Tampere University

Timo Tossavainen
Department of Arts, Communication and Education
Lulea University of Technology
SE-97187 Lulea


We first prove that any arithmetic subderivative of a rational number defines a function that is everywhere discontinuous in a very strong sense. Second, we show that although the restriction of this function to the set of integers is continuous (in the relative topology), it is not Lipschitz continuous. Third, we see that its restriction to a suitable infinite set is Lipschitz continuous. This follows from the solutions of certain arithmetic differential equations.

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(Concerned with sequences A000040 A003415.)

Received June 18 2019; revised version received October 9 2019. Published in Journal of Integer Sequences, October 15 2019.

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