A variation on arithmetic continuity

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Cakalli H.

Boletim da Sociedade Paranaense de Matematica, cilt.35, sa.3, ss.195-202, 2017 (Scopus)

Özet

A sequence (xk) of points in R, the set of real numbers, is called arithmetically convergent if for each ϵ > 0 there is an integer n such that for every integer m we have /xm - x/ < ϵ, where k/n means that k divides n or n is a multiple of k, and the symbol < m, n > denotes the greatest common divisor of the integers m and n. We prove that a subset of R is bounded if and only if it is arithmetically compact, where a subset E of R is arithmetically compact if any sequence of point in E has an arithmetically convergent subsequence. It turns out that the set of arithmetically continuous functions on an arithmetically compact subset of R coincides with the set of uniformly continuous functions where a function f defined on a subset E of R is arithmetically continuous if it preserves arithmetically convergent sequences, i.e., (f(xn) is arithmetically convergent whenever (xn) is an arithmetic convergent sequence of points in E.