Selective differentiation (Q1821887)

The purpose of this survey paper is to clarify some of the underlying principles and new ideas in the theory of the selective differentiation. In the fifth part of the paper, the author thoroughly discusses the problems of the relationships between selective, biselective, path system and composite differentiation. The first part of the paper deals with the history of the selective differentiation. There are mentioned some results and ideas of several papers which are basic for the introduction of the selective differentiation. In the second part, the author defines what is a selection and gives some examples of selections. The third part includes some interesting results about bilateral selective derivatives and the ideas of l-r-derivatives and biselective derivatives. The idea of the biselective differentiation has its origin in the Gleyzal's convergence idea of interval functions. The fourth part is devoted to the problems which are in connection with the fact that the selective derivative of any function (if it exists) is a function of the honorary Baire class two. The author deals with these problems in three research projects. In the first project he formulates the question: Whether it is possible for any selective differentiable function f with respect to a selection s, having the selective derivative sf' of the second class of Baire, to give an alternate selection t, such that the selective derivative tf' is of the first class of Baire. The answer to this question is in a common paper of the author and C. E. Weil. The second project deals with the question, whether it is possible to give some conditions on a selection s which guarantee that the selective derivative sf' of any function f (if it exists) is of the first class of Baire. This question was the motive for the introduction of the notion of balanced selections. The third project is in connection with Snyder's results on boundary functions and Stolz angle limit. The author investigates here the selective differentiation under so-called \(\alpha\) hv\(\beta\)-limit in the open half-plane. In the last part the author discusses some results of Lazarow and two papers which in between were published.