When introducing students to highly technical definitions for seemingly intuitive concepts (e.g., homotopy, continuity), how do you motivate the necessity of the definition? On the one hand, you would hope that the students are mathematically mature enough to appreciate a rigorous definition such as the epsilon-delta formulation of continuity. But if they are not (for example, epsilon-delta arises in their first truly proof-based class), are there standard cautionary tales that are especially convincing in conveying the worth of the technical definition?
(Context: I am teaching a course for students many of whom have not taken classes beyond linear algebra. The course serves as an introduction to proofs, and one part of the curriculum is continuity; for some of the students, this is the only place they will ever encounter epsilon-delta.)