The same issue comes up on teaching probability, and even worse, elementary stochastic processes. To do it correctly, you need measure theory, but that requires

The same issue comes up on teaching probability, and even worse, elementary stochastic processes. To do it correctly, you need measure theory, but that requires a whole course or courses. In fact, a lot of the results of freshman calculus require more mathematical analysis than most students really have command of. But we still teach it because the applications are so compelling, and the intuition is important to develop. Even more generally, as you probably know, there are tricky foundational issues in Mathematics that most professional mathematicians prefer not to pay attention to.