the manual systematizes the examples and counterexamples developed by the author to the basic definitions, theorems, rules, features and other provisions from the course of mathematical analysis. The manual will allow students of universities and technical universities to learn more deeply the content, meaning and applicability of the concepts and provisions of mathematical analysis and will make its teaching and study an interesting, creative process. Examples and counterexamples can be used by teachers in lectures and practical classes on mathematical analysis, as well as compose the content of a special course. For mathematics students, graduate students and university professors.