Improvable discontinuous function. (English) Zbl 0879.26006

If \(D\subset \mathbb R\) and \(f\: D\to \mathbb R\) is a bounded function, set \[ \begin{aligned} U(f)&=\big \{ x\in D:\;\lim_{t\to x}f(t) \not =f(x)\big \},\\ C(f)&=\big \{ x\in D:\;\lim_{t\to x}f(t) =f(x)\big \}.\\ \end{aligned} \] Let \(f_{(0)}=f\) and, for an ordinal number \(\alpha \), \(f_{(\alpha +1)}(x)=f_{(\alpha)}(x)\) if \(x\notin U(f_{(\alpha)})\) and \(f_{(\alpha +1)}(x)=\lim_{t\to x}f_{(\alpha)}(t)\) if \(x\in U(f_{(\alpha)})\). Moreover, let \(\mathcal A_{\alpha} =\big \{ f\: D\to \mathbb R: C(f_{(\alpha)})=D\big \}\). A function \(f\in \mathcal A_\alpha \backslash \bigcup_{0\leq \beta <\alpha}\mathcal A_{\beta}\) is called an \(\alpha\)-improvable discontinuous function. The author shows that \(\mathcal A_\alpha \backslash \bigcup_{0\leq \beta <\alpha} \mathcal A_\beta \not = \emptyset\) for any ordinal number \(\alpha <\omega_1\) and that \(\mathcal A_\alpha \not = \mathcal A_\beta\) if \(\alpha \not = \beta\). Furthermore, the author gives a necessary and sufficient condition under which a set \(A\subset D\) is the set of all points of continuity of some \(\alpha\)-improvable discontinuous function.
Reviewer: B.Opic (Praha)


26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable