On some issues of teaching mathematical analysis

In the article some methodical aspects of the teaching of mathematical analysis are discussed. This mathematical science is so big and important in the system of mathematical education that it is almost impossible to create an ideal course of lectures or an ideal book for all situations. There are always opportunities for improvement. In the process of the emergence and development of the Analysis, terminology and symbols developed. Sometimes this process was spontaneous and insufficiently thought out. It is difficult to make changes, but it is necessary. In modern conditions, there should be general mathematical terminology and symbols. Some points in this regard in the article are discussed. The author's proposals concern the terminology of convergence, the order of definition of the concept of a limit, the method of determining the limit and the order of its introduction, the terminology on the convergence of series, the introduction of the concept of the derivative, the differentiability of functions, partial derivatives, the generalization scheme for the concept of a definite integral, the order of studying topics in introduction to analysis, real numbers. The suggestions made are practically verified when reading lectures. The best students, comparing the lectures and books, agree with the author about these proposals. The article is intended for university teachers who read mathematical analysis and invites them to creative searches in this direction. Of course, every lecturer, in the concrete conditions, builds his vision in the framework of his duties, opportunities and methodological views and has the right to do so. This article is not a directive, but merely an invitation to discussion, since mathematical analysis gives a wide scope for creative searches and improvement of pedagogical skills.