The article presents the methodological approaches to the study of the continuity of functions, knowledge of which has a significant impact on both the mathematical and methodological training of future specialists. The formation of intuitive visual representations is a primary task for mastering the notion of continuity. As a result, Assessment of the level of mastering of concepts should be conducted, basically, not by the ability to reproduce the definitions of the continuity of a function at a point, but at the level of identification. In conclusion, we note that systems of computer mathematics should be used as ICT tools in training.