Real Analysis Exchange

Regulated Functions on Topological Spaces

Julie O’Donovan

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Abstract

A regulated function on the real line is a real valued function whose left-hand and right-hand limits exist at all points. In this paper we examine a generalization of regulated functions to functions defined on Davison Spaces, which are topological spaces with a little extra structure. Properties of such functions are discussed. Our main result concerns the set of discontinuities of these functions. We also prove that regulated functions defined on the natural numbers, with the cofinite topology, coincide with convergent sequences.

Article information

Source
Real Anal. Exchange Volume 33, Number 2 (2007), 405-416.

Dates
First available: 18 December 2008

Permanent link to this document
http://projecteuclid.org/euclid.rae/1229619418

Mathematical Reviews number (MathSciNet)
MR2458257

Zentralblatt MATH identifier
1158.26002

Subjects
Primary: 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) {For properties determined by Fourier coefficients, see 42A16; for those determined by approximation properties, see 41A25, 41A27} 54C35: Function spaces [See also 46Exx, 58D15]

Keywords
regulated functions discontinuous functions

Citation

O’Donovan, Julie. Regulated Functions on Topological Spaces. Real Analysis Exchange 33 (2007), no. 2, 405--416. http://projecteuclid.org/euclid.rae/1229619418.


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References

  • S. K. Berberian, Regulated functions: Bourbaki's alternative to the Riemann integral, Amer. Math. Monthly, 86(3) (1979), 208–211.
  • M. Brokate, P. Krejčí, Duality in the space of regulated functions and the play operator, Math. Z., 245(4) (2003), 667–688.
  • T. M. K. Davison, A generalization of regulated functions, Amer. Math. Monthly 86(3) (1979), 202–204.
  • J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York and London, 1963.
  • D. Fraňková, Regulated functions, Math. Bohem., 119 (1991), 20–59.
  • C. Goffman, G. Moran, D. Waterman, The structure of regulated functions, Proc. Amer. Math. Soc., 57(1) (1976), 61–65.
  • C. Goffman, D. Waterman, A characterisation of the class of functions whose Fourier series converge for every change of variable, London Math. Soc., 2(10) (1975), 69–74.
  • M. Grande, On the products of unilaterally continuous regulated functions, Real Anal. Exchange, 30(2) (2004/05), 871–876.
  • E. W. Hobson, The Theory of Functions of a Real Variable and the Theory of Fourier's Series, Cambridge University Press, Cambridge, 1907.
  • P. Krejčí, P. Laurençot, Hysteresis filtering in the space of bounded measurable functions, Boll. Unione Mat. Ital., 5-B (2002), 755–772.
  • T. Mikosch, R. Norvaiša, Stochastic integral equations without probability, Bernoulli, 6 (2000), 401–434.
  • G. Moran, Of regulated and steplike functions, Trans. Amer. Math. Soc., 231(1) (1977), 249–257.
  • R. Norvaiša, The p-variation and an extension of the class of semimartingales, Acta Appl. Math., 79(1-2), (2003).
  • P. Pierce, D. Velleman, Some generalizations of the notion of bounded variation, Amer. Math. Monthly, 113(10) (2006), 897–904.
  • M. Tvrdý, Differential and integral equations in the space of regulated functions, Mem. Differential Equations Math. Phys., 25 (2002), 1–104.
  • A. I. Vol'pert, S. I. Khudyaev, Analysis in Classes of Discontinuous Functions and Equations of Mathematical Physics, Martinus Nijhoff Publishers, Dordrecht, 1985.