Real functions.

*(English)*Zbl 0581.26001
Lecture Notes in Mathematics. 1170. Berlin etc.: Springer-Verlag. VII, 229 p. DM 31.50 (1985).

The monograph deals with questions which are in relation to the continuity and to the differentiation of real functions of a real variable, in certain generalized senses. The author’s intention was to make accessible some results of the relatively large literature about these questions which are known only to specialists also to any interested mathematician. This is possible, since ideas and techniques are natural and simple and they do not require any special knowledge only the usual basics of real analysis. Part of the material of this monograph the author presented in a series of seminars at the University of California at Santa Barbara in the spring 1984 during the special year in Real Analysis that was held there.

Several results, that concern the usual notions of the continuity and the differentiation of real functions of a real variable, appear in analogical formulations also using generalized notions of the continuity and the differentiation. The author’s intention was to express several of them by the author’s constructed notion of a general structure - a local system of sets. By a local system of sets in the set R of all real numbers, the author means an indexed system of nonempty systems of subsets of R-\(\{{\mathcal S}(x):x\in R\}\)- indexed by R and fulfilling the following properties: For all \(x\in R\) it holds: \((i)\quad \{x\}\not\in {\mathcal S}(x);\) (ii) if \(S\in {\mathcal S}(x),\) then \(x\in S;\) (iii) if \(S_ 1\in {\mathcal S}(x)\) and \(S_ 1\subset S_ 2\subset R,\) then \(S_ 2\in {\mathcal S}(x)\) and (iv) if \(S\in {\mathcal S}(x)\) and \(\delta >0,\) then \(S\cap (x-\delta,x+\delta)\in {\mathcal S}(x).\) The notions of the limit, of the differentiation, of the covers, etc. are expressed by local system of sets.

The monograph has seven chapters entitled by: local system, cluster sets, continuity, variation of a function, monotonicity, relations among derivatives and the Denjoy-Young relations and appendix: set porosity.

The material in the monograph is good organized, the explanation is interesting and several proofs and reflections that do not require some special ideas are left to the reader. There are also much references on directly or indirectly related papers. I think that the author was successful in writing this book, so that he fulfilled his intention.

Several results, that concern the usual notions of the continuity and the differentiation of real functions of a real variable, appear in analogical formulations also using generalized notions of the continuity and the differentiation. The author’s intention was to express several of them by the author’s constructed notion of a general structure - a local system of sets. By a local system of sets in the set R of all real numbers, the author means an indexed system of nonempty systems of subsets of R-\(\{{\mathcal S}(x):x\in R\}\)- indexed by R and fulfilling the following properties: For all \(x\in R\) it holds: \((i)\quad \{x\}\not\in {\mathcal S}(x);\) (ii) if \(S\in {\mathcal S}(x),\) then \(x\in S;\) (iii) if \(S_ 1\in {\mathcal S}(x)\) and \(S_ 1\subset S_ 2\subset R,\) then \(S_ 2\in {\mathcal S}(x)\) and (iv) if \(S\in {\mathcal S}(x)\) and \(\delta >0,\) then \(S\cap (x-\delta,x+\delta)\in {\mathcal S}(x).\) The notions of the limit, of the differentiation, of the covers, etc. are expressed by local system of sets.

The monograph has seven chapters entitled by: local system, cluster sets, continuity, variation of a function, monotonicity, relations among derivatives and the Denjoy-Young relations and appendix: set porosity.

The material in the monograph is good organized, the explanation is interesting and several proofs and reflections that do not require some special ideas are left to the reader. There are also much references on directly or indirectly related papers. I think that the author was successful in writing this book, so that he fulfilled his intention.

Reviewer: L.Mišik

##### MSC:

26-02 | Research exposition (monographs, survey articles) pertaining to real 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 |