Arithmetic Subderivatives: Discontinuity and Continuity
Pentti Haukkanen and Jorma K. Merikoski
Faculty of Information Technology and Communication Sciences
FI-33014 Tampere University
Department of Arts, Communication and Education
Lulea University of Technology
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.
Full version: pdf,
(Concerned with sequences
Received June 18 2019; revised version received October 9 2019.
Published in Journal of Integer Sequences,
October 15 2019.
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