On Regulated Functions

The main purpose of this review article is to present the concept of a regulated function and to indicate the connection of the class of regulated functions with other significant classes of functions. In particular, we give a characterization of regulated functions in terms of step functions and we show that the linear space of regulated functions form a Banach space under the classical supremum norm. 

Full text (pdf)

1. J. Appell, J. Banaś and N. Merentes, Bounded Variation and Around, Series in Nonlinear Analysis and Applications 17, De Gruyter, Berlin 2014.
2. G. Aumann, Reelle Funktionen, Springer Verlag, Berlin 1954.
3. J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York 1969.
4. D. Fraňková, Regulated functions, Math. Bohemica 116 (1991) 20-59.
5. C. Goffman, G. Moran and D. Waterman, The structure of regulated functions, Proc. Amer. Math. Soc. 57 (1976) 61-65.
6. R. Łochowski, R. Ghomrasni, The play operator, the truncated variation and the generalization of the Jordan decomposition, Math. Appl. Sci. 38 (2015) 403-419.
7. S. Łojasiewicz, An Introduction to the Theory of Real Functions, J. Wiley and Sons, Chichester 1988.
8. F. Prus-Wiśniowski, Some remarks on functions of bounded φ-variation, Ann. Soc. Math. Pol., Comment. Math. 30 (1990) 147-166.
9. F. Prus-Wiśniowski, Continuous functions of bounded φ-variation, Ann. Soc. Math. Pol., Comment. Math. 31 (1991) 127-146.
10. W. Rudin, Principles of Mathematical Analysis, McGraw Hill, New York 1964.
11. R. Sikorski, Funkcje Rzeczywiste, PWN, Warsaw 1958 (in Polish).