Abstract and Applied Analysis

Slowly Oscillating Continuity

H. Çakalli

Abstract

A function $f$ is continuous if and only if, for each point ${x}_{0}$ in the domain, ${\lim{}}_{n{\rightarrow}\infty{}}f({x}_{n})=f({x}_{0})$, whenever ${\lim{}}_{n{\rightarrow}\infty{}}{x}_{n}={x}_{0}$. This is equivalent to the statement that $(f({x}_{n}))$ is a convergent sequence whenever $({x}_{n})$ is convergent. The concept of slowly oscillating continuity is defined in the sense that a function $f$ is slowly oscillating continuous if it transforms slowly oscillating sequences to slowly oscillating sequences, that is, $(f({x}_{n}))$ is slowly oscillating whenever $({x}_{n})$ is slowly oscillating. A sequence $({x}_{n})$ of points in $\mathbf{R}$ is slowly oscillating if ${\lim{}}_{\lambda{}{\rightarrow}{1}^{+}}{{\stackrel{\rule{10pt}{1pt}}{\lim{}}}_{n}{\max{}}_{n+1\leq{}k\leq{}[\lambda{}n]}}_{}|{x}_{k}-{x}_{n}|=0$, where $[\lambda{}n]$ denotes the integer part of $\lambda{}n$. Using $\varepsilon{}>0$'s and $\delta{}$'s, this is equivalent to the case when, for any given $\varepsilon{}>0$, there exist $\delta{}=\delta{}(\varepsilon{})>0$ and $N=N(\varepsilon{})$ such that $|{x}_{m}-{x}_{n}|< \varepsilon{}$ if $n\geq{}N(\varepsilon{})$ and $n\leq{}m\leq{}(1+\delta{})n$. A new type compactness is also defined and some new results related to compactness are obtained.

Article information

Source
Abstr. Appl. Anal., Volume 2008 (2008), Article ID 485706, 5 pages.

Dates
First available in Project Euclid: 9 September 2008

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1220969159

Digital Object Identifier
doi:10.1155/2008/485706

Mathematical Reviews number (MathSciNet)
MR2393124

Zentralblatt MATH identifier
1153.26002

Citation

Çakalli, H. Slowly Oscillating Continuity. Abstr. Appl. Anal. 2008 (2008), Article ID 485706, 5 pages. doi:10.1155/2008/485706. https://projecteuclid.org/euclid.aaa/1220969159

References

• H. Fast, “Sur la convergence statistique,” Colloquium Mathematicum, vol. 2, pp. 241–244, 1951.
• J. A. Fridy, “On statistical convergence,” Analysis, vol. 5, no. 4, pp. 301–313, 1985.
• A. Zygmund, Trigonometric Series. Vol. II, Cambridge University Press, New York, NY, USA, 2nd edition, 1959.
• H. Robbins and R. C. Buck, “Advanced problems and solutions: solutions: 4216,” The American Mathematical Monthly, vol. 55, no. 1, p. 36, 1948.
• E. C. Posner, “Summability-preserving functions,” Proceedings of the American Mathematical Society, vol. 12, no. 1, pp. 73–76, 1961.
• T. B. Iwiński, “Some remarks on Toeplitz methods and continuity,” Commentationes Mathematicae. Prace Matematyczne, vol. 16, pp. 37–43, 1972.
• V. K. Srinivasan, “An equivalent condition for the continuity of a function,” The Texas Journal of Science, vol. 32, no. 2, pp. 176–177, 1980.
• J. Antoni, “On the $A$-continuity of real function. II,” Mathematica Slovaca, vol. 36, no. 3, pp. 283–288, 1986.
• J. Antoni and T. Šalát, “On the $A$-continuity of real functions,” Universitas Comeniana Acta Mathematica Universitatis Comenianae, vol. 39, pp. 159–164, 1980.
• E. Spigel and N. Krupnik, “On $A$-continuity of real functions,” Journal of Analysis, vol. 2, pp. 145–155, 1994.
• E. Öztürk, “On almost-continuity and almost $A$-continuity of real functions,” Université d'Ankara. Faculté des Sciences. Communications. Série A1. Mathématiques, vol. 32, no. 4, pp. 25–30, 1983.
• E. Savaş and G. Das, “On the $A$-continuity of real functions,” \.Istanbul Üniversitesi. Fen Fakültesi. Matematik Dergisi, vol. 53, pp. 61–66, 1994.
• J. Borsík and T. Šalát, “On $F$-continuity of real functions,” Tatra Mountains Mathematical Publications, vol. 2, pp. 37–42, 1993.
• J. Connor and K.-G. Grosse-Erdmann, “Sequential definitions of continuity for real functions,” The Rocky Mountain Journal of Mathematics, vol. 33, no. 1, pp. 93–121, 2003.
• H. Çakalli, “Sequential definitions of compactness,” Applied Mathematics Letters, vol. 21, no. 6,pp. 594–598, 2008.