## 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)

### MSC:

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

### Keywords:

improvable discontinuous function