Regulated functions and integrability

Ján Gunčaga

Abstract


Properties of functions defined on a bounded closed interval, weaker than continuity, have been considered by many mathematicians. Functions having both sides limits at each point are called regulated and were considered by J. Dieudonné [2], D. Fraňková [3] and others (see for example S. Banach [1], S. Saks [8]). The main class of functions we deal with consists of piece-wise constant ones. These functions play a fundamental role in the integration theory which had been developed by Igor Kluvanek (see Š. Tkacik [9]). We present an outline of this theory.

Keywords


regulated function, piece-wise constant function

Mathematics Subject Classification


26A15, 26A39

References


Banach, S. "On measures in independent field." Stud. Mat. 10 (1948): 159-177.

Dieudonné, J. "Foundations of Modern Analysis." New York-London: Academic Press, 1969.

Franková, D. "Regulated Functions." Math. Bohem. 116 (1991): 20-59.

I. Kluvánek. "Manuscripts of the differential and integral calculus."

Königsberger, B. "Analysis I." Berlin-Heidelberg-New York: Springer Verlag, 2001.

Mityushev, V.V. and S.V. Rogosin. "Constructive methods for boundary value problems for analytic functions." Progress in analysis 2. River Edge, NJ: World Sci. Publ., 2003. 769-777.

Pfeffer, W.F. "Integrals and Measures." New York--Basel: Marcel Dekker, 1977.

Saks, S. "Theory of the Integral." New York: Dover Publications, 1964.

Tkacik, Š. "Spojitosta limity trochu inak." Zbornik Konferencie Setkání Kateder Matematiky Ceské a Slovenské Republiky pripravující budoucí ucitele, Ústí nad Labem, 2004. 85-89.


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