[in9]

> I think math exists to solve real problems, so we should give examples of solving real problems.

This is a view that, depending on the lecturer, can be the total opposite of their view. See, many mathematicians don't have this instrumentalist view on math and some even despise it with their heart. They don't care about applications and don't get particularly excited when a problem was solved in another domain with their technique.

Also, solving a particular problem is that, a particularity. They look for generalizations and for rules and methods that can be applied regardless of the domain. That is why weird and contrived examples are the bulk of their logic, since when you look for generalizations, those are the cases where it breaks. Does continuity implies differentiability? For almost all practical cases yes, but when you want to be general, thats when you get the Weierstrass function [1], a weird function that needs to be studied if you want to develop a theory of real valued continuous functions. Yet, such knowledge is extremely impractical for you average engineer or deep learning researcher.

And ok, you might argue that "I'm not at such level, I don't need those particular theoretically interesting cases since I'm not going to study advanced mathematics at that level". Ok fine, I respect that. But that is not part of the "culture" of mathematics (a culture which your lecturer likely comes from). Since the subject is learned and investigated with this "quest for generality which makes us study weird examples" this is how it is passed along, and this is how it is studied and structured.

All that to conclude: I think its best to respect how math is structured. First, it is precisely the type of thought that I think very other applied fields lack, but that actually helps solving problems. It is by seeking generality, and looking at corner cases that stuff gets built. And mathematicians are the best at formalizing those because this is how they advance their craft. Second, and last, its by creating your own investigation and development of examples that you can stride for a more applied and practical version of the course. That way you'll get the most out of the course.

[1]: https://en.wikipedia.org/wiki/Weierstrass_function

[eigenket]

> Does continuity implies differentiability? For almost all practical cases yes, but when you want to be general, thats when you get the Weierstrass function [1], a weird function

Ok I have to jump in and disagree with you. Non-differential continuous functions are far more ubiquitous (and useful) than you're suggesting here. The most obvious examples are the absolute value function and ReLU (rectified linear unit) activation function which turns up in a machine learning/neural network context.

I think you're thinking about being non-differentiable everywhere, but it's very easy to cook up examples of practically relevant functions which turn up to be non-differentiable somewhere.