A Note on Statistical Continuity of Functions

Year 2024, Volume: 7 Issue: 4, 212 - 217, 31.12.2024

İsmail Osmanoğlu , Erdinç Dündar

https://doi.org/10.33401/fujma.1443574

Abstract

In the present paper, we first recall the notion of statistical convergence of double sequences defined on topological spaces and reduced equivalent to the definition, along with some of its basic properties. Later, we define the concept of statistically continuous as a general case of the continuous function using the statistical convergence of double sequences. We define strong and weak statistically continuous functions as final definitions that arise as a direct consequence of statistically continuous functions. In the rest of the paper, we analyze the implications between the given definitions and investigate additional conditions for equality.

Keywords

Statistical continuity, Statistical convergent double sequence, Statistical convergent function

References

• [1] A. Zygmund, Trigonometric Series, Cambridge University Press, Cambridge, (2002). $ \href{https://doi.org/10.1017/CBO9781316036587}{\mbox{[CrossRef]}} $
• [2] H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241-244. $ \href{http://matwbn.icm.edu.pl/ksiazki/cm/cm2/cm2137.pdf}{\mbox{[Web]}} $
• [3] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., 2 (1951), 73-74.
• [4] I.J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66 (1959), 361-375. $ \href{https://doi.org/10.1080/00029890.1959.11989303}{\mbox{[CrossRef]}} $
• [5] R.C. Buck, Generalized asymptotic density, Am. J. Math., 75(2) (1953), 335-346. $\href{https://doi.org/10.2307/2372456}{\mbox{[CrossRef]}} $
• [6] G. Di Maio and D.R. Kocinac, Statistical convergence in topology, Topol. Appl., 156(1) (2008), 28-45. $\href{https://doi.org/10.1016/j.topol.2008.01.015}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-53949096545&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Statistical+convergence+in+topology%22%29&sessionSearchId=3e756a729286e885f51153c1b53cbc63&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000260957900006}{\mbox{[Web of Science]}} $
• [7] M. Mursaleen and O.H.H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl., 288(1) (2003), 223-231. $ \href{https://doi.org/10.1016/j.jmaa.2003.08.004}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-0346273049&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Statistical+convergence+of+double+sequences%22%29&sessionSearchId=3e756a729286e885f51153c1b53cbc63&relpos=79}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000187210100018}{\mbox{[Web of Science]}} $
• [8] V. Renukadevi and P. Vijayashanthi, Statistical convergence of double sequences, Jordan J. Math. Stat., 14(4) (2021), 787-808. $ \href{https://doi.org/10.47013/14.4.12}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85123838655&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Statistical+convergence+of+double+sequences%22%29&sessionSearchId=3e756a729286e885f51153c1b53cbc63&relpos=17}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000731919900012}{\mbox{[Web of Science]}} $
• [9] H. Çakallı, A new approach to statistically quasi Cauchy sequences, Maltepe J. Math., 1(1) (2019), 1-8. $\href{https://dergipark.org.tr/en/download/article-file/565465}{\mbox{[Web]}} $
• [10] A. Caserta, G. Di Maio and L.D.R. Kocinac, Statistical convergence in function spaces, Abstr. Appl. Anal., 2011(1)(2011). $ \href{https://doi.org/10.1155/2011/420419}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84855549781&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Statistical+convergence+in+function+spaces%22%29&sessionSearchId=3e756a729286e885f51153c1b53cbc63&relpos=1}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000298684200001}{\mbox{[Web of Science]}} $
• [11] H.Ş. Kandemir, On I-deferred statistical convergence in topological groups, Maltepe J. Math., 1(2) (2019), 48-55. $\href{https://dergipark.org.tr/en/download/article-file/843195}{\mbox{[Web]}} $
• [12] F. Nuray, E. Dündar and U. Ulusu, Some generalized definitions of uniform continuity for real valued functions, Creat. Math. Inform., 29(2) (2020), 165-170. $ \href{https://doi.org/10.37193/CMI.2020.02.08}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85208221814&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Some+generalized+definitions+of+uniform+continuity+for+real+valued+functions%22%29&sessionSearchId=3e756a729286e885f51153c1b53cbc63&relpos=0}{\mbox{[Scopus]}} $
• [13] F. Nuray, E. Dündar and U. Ulusu, Wijsman statistical convergence of double sequences of set, Iran. J. Math. Sci. Inform., 16(1) (2021), 55-64. $ \href{https://doi.org/10.29252/ijmsi.16.1.55}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85105621730&origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001097286800004}{\mbox{[Web of Science]}} $
• [14] R.F. Patterson and H. Çakallı, Quasi Cauchy double sequences, Tbilisi Math. J., 8(2) (2015), 211-219. $ \href{https://doi.org/10.1515/tmj-2015-0023}{\mbox{[CrossRef]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000219434000018}{\mbox{[Web of Science]}} $
• [15] U. Ulusu, F. Nuray and E. Dündar, $ \mathfrak{I} $-limit and $ \mathfrak{I} $-cluster points for functions defined on amenable semigroups. Fundam. J. Math. Appl., 4(2) (2021), 45-48. $ \href{https://doi.org/10.33401/fujma.842104}{\mbox{[CrossRef]}} $
• [16] E. Gülle and U. Ulusu, Wijsman deferred invariant statistical and strong $ p $-deferred invariant equivalence of order $ \alpha $, Fundam. J. Math. Appl., 6(4) (2023), 211-217. $ \href{https://doi.org/10.33401/fujma.1364368}{\mbox{[CrossRef]}} $
• [17] Ö. Kişi and E. Güler, $ \mathcal{I}$-Cesaro summability of a sequence of order $ \alpha $ of random variables in probability, Fundam. J.Math. Appl., 1(2) (2018), 157-161. $ \href{https://doi.org/10.33401/fujma.480808}{\mbox{[CrossRef]}}$
• [18] S. Erdem and S. Demiriz, A study on strongly almost convergent and strongly almost null binomial double sequence spaces. Fundam. J. Math. Appl., 4(4) (2021), 271-279. $ \href{https://doi.org/10.33401/fujma.987981}{\mbox{[CrossRef]}} $
• [19] M. Candan, A new aspect for some sequence spaces derived using the domain of the matrix $ \hat{\hat{B}} $, Fundam. J. Math. Appl., 5(1) (2022), 51-62. $\href{https://doi.org/10.33401/fujma.1003752}{\mbox{[CrossRef]}} $
• [20] F. Gökçe, Compact and matrix operators on the space $\left\vert \overline N_p^{\phi }\right\vert _{k}$, Fundam. J. Math. Appl., 4(2) (2021), 124-133. $\href{https://doi.org/10.33401/fujma.882309}{\mbox{[CrossRef]}} $
• [21] S. Aydın and H. Polat, Difference sequence spaces derived by using Pascal transform. Fundam. J. Math. Appl., 2(1) (2019), 56-62. $ \href{https://doi.org/10.33401/fujma.541721}{\mbox{[CrossRef]}} $
• [22] H. Çakallı , On G-continuity, Comput. Math. Appl., 61(2) (2011), 313-318. $ \href{https://doi.org/10.1016/j.camwa.2010.11.00}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-78651228578&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On++G+-continuity%22%29&sessionSearchId=3e756a729286e885f51153c1b53cbc63&relpos=1}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000287553400016}{\mbox{[Web of Science]}} $
• [23] T. Salat, On statistically convergent sequences of real numbers, Math. Slovaca 30(2) (1980), 139-150. $ \href{https://dml.cz/handle/10338.dmlcz/136236}{\mbox{[Web]}} $
• [24] J.A. Fridy, On statistical convergence, Anal., 5(4) (1985), 301-313. $ \href{https://doi.org/10.1524/anly.1985.5.4.301}{\mbox{[CrossRef]}} $
• [25] H. Çakallı and M. K. Khan, Summability in topological spaces, Appl. Math. Lett., 24(3) (2011), 348-352. $ \href{https://doi.org/10.1016/j.aml.2010.10.021}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-78649991658&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Summability+in+topological+spaces%22%29&sessionSearchId=3e756a729286e885f51153c1b53cbc63&relpos=2}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000285659800022}{\mbox{[Web of Science]}} $
• [26] A. Pringsheim, Zur Theorie der zweifach unendlichen zahlenfolgen, Math. Ann., 53(3) (1900), 289-321. $\href{https://doi.org/10.1007/BF01448977}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-0346408596&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Zur+Theorie+der+zweifach+unendlichen+Zahlenfolgen%22%29&sessionSearchId=3e756a729286e885f51153c1b53cbc63&relpos=0}{\mbox{[Scopus]}} $