Some typical results on bounded Baire 1 functions (Q795188)

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 \({\mathcal F}\).