## On some properties of the class $$\mathcal A^\star$$.(English)Zbl 1024.26001

For $$x,y\in I=[0,1]$$ let $$I_{(x,y)}$$ denote the closed interval with the ends $$x$$ and $$y$$. For a function $$f\colon I\to I$$ and for $$x,y\in I$$ let $$A^f_{xy}= I_{(f(x),f(y))}\setminus f(I_{(x,y)})$$ and $$A^f=\bigcup_{x,y\in I}A^f_{xy}$$. Let $$\mathcal J$$ be the family of all $$\sigma$$-ideals of boundary subsets of $$I$$ which contain all singletons. For $$J\in {\mathcal J}$$ let $${\mathcal A}^{\star}_{J}$$ denote the family of all nowhere constant functions $$f\colon I\to I$$ with $$A^f\in J$$. Then let $${\mathcal A}^{\star}=\bigcup_{J\in {\mathcal J}}{\mathcal A}^{\star}_{J}$$. Moreover, for $$J\in {\mathcal J}$$ let $${\mathcal Q}_{\mathcal D}^{J}$$ denote the family of all nowhere constant functions $$f\colon I\to I$$ such that $$A^f_{xy}\in J$$ for every $$x,y\in I$$. Then let $${\mathcal Q}_{\mathcal D}=\bigcup_{J\in{\mathcal J}}{\mathcal Q}_{\mathcal D}^{J}$$. It is easy to observe that $${\mathcal A}^{\star}\subset {\mathcal Q}_{\mathcal D}$$. The main result of the first part of the paper is the equality $${\mathcal A}^{\star}={\mathcal Q}_{\mathcal D}$$. In the second part of the paper the authors work with the space of real-valued functions defined on the space $${\mathcal A}^{\star}$$. The results are connected with the class $${\mathcal D}_{\mathcal P}$$ of all functions $$f\colon {\mathcal A}^{\star}\to {\mathbb{R}}$$ such that $$f(L)$$ is a connected set for each arc $$L\subset {\mathcal A}^{\star}$$. It is proven that: (1) There exists $$\Phi\subset {\mathcal A}^{\star}$$ such that if $$f\in{\mathcal D}_{\mathcal P}$$ and $$f|\Phi$$ or $$f|({\mathcal A}^{\star}\setminus\Phi)$$ is quasi continuous (cliquish), $$f$$ is so (but the corresponding assertion for continuity is false). (2) The set $${\mathcal D}_{\mathcal P}$$ is porous at each point of some big subset of the space of all functions $$f\colon {\mathcal A}^{\star}\to{\mathbb{R}}$$ (with the metric of uniform convergence).

### MSC:

 26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable 26A21 Classification of real functions; Baire classification of sets and functions 26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems 54C30 Real-valued functions in general topology