Many believe (I think rightly so) that the presentation of counterexamples should play an important role in the teaching upper level mathematics courses such as real analysis and topology. Counterexamples show why the hypotheses of various theorems are important. Further, counterexamples very often add to students' intuition and ability to quickly recognize false propositions.
Some examples of counterexamples that could be provided in a first year calculus course include:
- Removing or varying hypotheses in L'Hopital's Rule,
- Removing or varying hypotheses in Intermediate Value Theorem
- Converse of Differentiability implies continuity.
Should counterexamples play an important role in a first year calculus course taken largely by non-majors?