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### On the maxima of two functions which are simultaneously unilaterally approximately continuous and approximately regulated

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In this article we investigate the maxima of two unilaterally approximately continuous and approximately regulated functions. In particular we prove that if / is the maximum of two unilaterally approximately continuous and approximately regulated functions then for each x is an element of Dunap(f) = {x : f is not unilaterally approximately continuous at x} the inequality f(x) < max(fap(x+),fap(x-)) holds. Moreover, we show some condition ensuring that an approximately regulated function f such that Dap(f) is countable and for each x is an element of Dunap(f) the inequality f(x) < max(fap(x+),fap(x-)) holds, is the maximum of two unilaterally approximately continuous and approximately regulated functions.
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Strony
523--531
Opis fizyczny
Bibliogr. 6 poz.
Twórcy
autor
• Institute of Mathematics, Bydgoszcz Academy, Plac Weyssenhoffa 11, 85-072 Bydgoszcz, Poland
Bibliografia
• [1] A. M. Bruckner, Differentiation of Real Functions, Lectures Notes in Mathematics 659, Springer-Verlag, Berlin Heidelberg New York 1978.
• [2] M. Grande, On the sums of unilaterally approximately continuous and approximate jump functions, Real Analysis Exchange Vol. 28(2), 2002/2003, pp. 1-8.
• [3] M. Grande, On the maximums of unilaterally continuous regulated functions, accepted to Real Analysis Exchange Vol. 29(2), 2004.
• [4] C. S. Reed, Pointwise limits of sequences of functions, Fundamenta Math. 67 (1970), 183-193.
• [5] R. Sikorski, Funkcje Rzeczywiste I, Warszawa 1958 (in Polish).
• [6] F. D. Tall, The density topology, Pacific J. Math. 62 (1976), 275-284.
Typ dokumentu
Bibliografia