A variation on arithmetic continuity
Küçük Resim Yok
Tarih
2017
Yazarlar
Dergi Başlığı
Dergi ISSN
Cilt Başlığı
Yayıncı
SOC PARANAENSE MATEMATICA
Erişim Hakkı
info:eu-repo/semantics/openAccess
Özet
A sequence (x(k)) of points R, the set of real numbers, is called arithmetically convergent if for each epsilon > 0 there is an lat for every integer m, we have vertical bar x(m) - x(<m,n>)vertical bar < epsilon, where k vertical bar 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 lit is arithmetically continuous if it preserves arithmetically convergent sequences, i.e., (f (x(n)) is arithmetically convergent whenever (x(n)) is an arithmetic convergent sequence of points in E.
Açıklama
Anahtar Kelimeler
arithmetical convergent sequences, boundedness, uniform continuity
Kaynak
BOLETIM SOCIEDADE PARANAENSE DE MATEMATICA
WoS Q Değeri
N/A
Scopus Q Değeri
Q3
Cilt
35
Sayı
3
