Ahmad Nusrat Mammadov , Ramiz Memmedjafar Beyaliyev

Abstract. The presented work is devoted to materials on the use of counterexamples, which play an important role in increasing the effectiveness of teaching elements of Mathematical Analysis in the course “Linear Algebra and Mathematical Analysis” in technical and economic higher education institutions. The counterexamples mainly cover the part of the Mathematical Analysis course related to functions of one variable (up to the elements of integral calculus). The materials on Mathematical Analysis reflect issues related to constructing examples (counterexamples) that demonstrate whether the converse of a theorem or another statement is true, that a statement is not valid when at least one of its conditions is not satisfied, and that certain conditions may be sufficient (necessary) but not necessary (sufficient). All of this plays an important role in helping students master at a high level other mathematical subjects, as well as disciplines that include elements of Mathematical Analysis (“Econometrics,” “Marketing Management,” “Macroeconomics,” “Mathematical Statistics,” “Probability Theory,” etc.). It should also be noted that the use of counterexamples is considered beneficial not only in all topics of Mathematical Analysis, but also in the teaching of such subjects as “Linear Algebra,” “Analytic Geometry,” “Differential Equations,” and others. It contributes to the development of students’ independent working skills, logical thinking, and enhancement of their creative abilities.

Key words: counterexample, one-sided limits, one-sided derivatives, indeterminate forms, Rolle’s theorem, Lagrange’s theorem, Cauchy’s theorem, L’Hôpital–Bernoulli rules, Dirichlet function.