О НЕКОТОРЫХ ПОНЯТИЯХ И ТЕОРЕМАХ МАТЕМАТИЧЕСКОГО АНАЛИЗА

Abstract

The article presents the results obtained by the author in search of innovative forms and methods of presenting educational and methodological information in order to achieve a better understanding, assimilation and consolidation of the educational material read by students, and at the output – the ability to apply the knowledge gained in practice. Based on the axiom of continuity of the set of real numbers, definitions of a continuous set, a discontinuous set with a jump and with a gap are given. As well as the examples. The use of symbolism and the laws of mathematical logic, as also the equivalence of the necessary condition for the existence of a mathematical concept and a sufficient condition for the non-existence of this concept, allowed us to obtain a number of significant results with regard to the continuity of the function, the local extremum of the function and the inflection point of the function graph in the Roll theorem. The article shows the existence of an alternative to the standard form of presentation of the necessary condition for the continuity of a function at a point and the necessary condition for the continuity of a function on a set. A new criterion for the continuity of a function on a set is presented, which greatly simplifies the study of a function for continuity on a set. The consequence of the criterion was a new definition of the continuity of a function on a set. Heaviside and Dirichlet functions are given as an example. Some concepts and theorems set forth in the article are used by the author in classes on mathematical analysis.