When discussing Functions, we also looked at continuity issues, unfortunately not as much as the topic deserves but it is an exciting topic. Here is something about the catastrophe machine. There had been a time, when the excitement about lack of continuity was raging in the form of Catastrophe theory. It had been one of the fancy theories in mathematics with lots of applications. It had been Fashion (PDF). There are simple models which explain how discontinuities appear Example. It makes a nice lecture also in single variable calculus. Here are two pages from the book of V.I. Arnold called "Catastrophe theory". It explains a bit why scientists were so excited about catastrophes once: (It is still ok to be excited now!)

There would be a lot to say about the culture of math. About 25 years ago for example, an argued flared up about how much rigor is needed in math: [ a paper from 1993 , and som responses by Mathematicians We see such discussions again and again also in education: how rigorous should material be presented? Do we even want to cover topics like continuity in calculus courses? If yes, how much should be proven?