You can build a much more abstract model of mathematical analysis by starting with open and closed sets instead of ye olde derivations and integrals. This chang

You can build a much more abstract model of mathematical analysis by starting with open and closed sets instead of ye olde derivations and integrals. This changes the balance somewhat; instead of defining continuity of a function through limits, you say that a continuous function is a function for which preimage of every open set is open. AFAIK this isn't taught often, especially not to not-mathematicians by trade. The tradition is to start with infinetesimal calculus, Newton and Leibniz style, which also produces much more "tangible" effects (you can solve some equations and tasks like that), plus can be easily tested in a written exam. Here in CZ, one guy (Petr Vopěnka) spent a lot of time on trying to build mathematical models in this abstract way; his work included set theory and analysis among others. He is highly regarded, but no one adapted his way of teaching AFAIK.