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Some typical results on bounded Baire 1 functions. (English) Zbl 0542.26004

In the paper the authors deal with functions f:[0,1]\(\to R\) (R - the real line). Let b\({\mathcal A}\), \(b\Delta\), b\({\mathcal D}{\mathcal B}^ 1\), b\({\mathcal B}^ 1\) be Banach spaces of bounded approximately continuous functions, bounded derivatives, bounded Daboux Baire 1 functions and bounded Baire 1 functions, resp., with norm \(\| f\| =\sup | f|.\) Some typical properties of functions of the above spaces are given. These concern of level sets, of continuity points and of the range. Let us introduce some of the reached results: Theorem 1.10. The set of functions f such that \(f^{-1}(y)\) is a nowhere dense (Lebesgue) nullset for all \(y\in R\) is a residual \(G_{\delta}\) set in any of b\({\mathcal A}\), \(b\Delta\) and b\({\mathcal D}{\mathcal B}^ 1\). Theorem 2.4. Let \(\mu\) be an arbitrary finite Borel measure on [0,1] and \({\mathcal F}=b{\mathcal A},\quad b\Delta,\quad b{\mathcal D}{\mathcal B}^ 1,\quad b{\mathcal B}^ 1.\) Then \(\{f\in {\mathcal F}:\mu(C_ f)=0\}\) is an everywhere dense \(G_{\delta}\) set in \({\mathcal F} (C_ f\)- the set of continuity points of f). Theorem 3.5. Let \({\mathcal F}=b{\mathcal A},\quad b\Delta,\quad b{\mathcal D}{\mathcal B}^ 1,\quad b{\mathcal B}^ 1.\) Then the family \(\{f\in {\mathcal F}: f(C_ f)\) is of power of continuum} is an everywhere dense \(G_{\delta}\) set in \({\mathcal F}\).
Reviewer: P.Kostyrko

MSC:

26A21 Classification of real functions; Baire classification of sets and functions
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
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[1] A. M. Bruckner,Differentiation of Real Functions, Lecture Notes in Math., 659, Springer, 1978. · Zbl 0382.26002
[2] J. Ceder, G. Petruska, Most Darboux Baire 1 functions map big sets onto small,Acta Math. Hung.,41 (1983), 37–46. · Zbl 0525.26004 · doi:10.1007/BF01994059
[3] Z. Zahorski, Sur la premiere dérivée,Trans. Amer. Math. Soc.,69 (1950), 1–54. · Zbl 0038.20602
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