Detailed solutions to problems and answers are given, methodological instructions are given, typical errors are analyzed and relevant counterexamples are given, the necessary statements are formulated, and sometimes proved, taken, as a rule, from lectures on mathematical analysis given to students of the 1st year of MIPT. Solutions to some problems are given in several ways, set out in one or different versions. Whole!" manuals to assist the student in mastering the methods of solving problems, which is necessary for the successful completion of the examination work.