An interesting approach to calculus. Instead of using the limit-based definition of continuity, Olver defines a function to be continuous if the inverse image of every open set is open (as in topology). He says this is both more rigorous and more understandable for beginners.
You can check out Olver’s approach here [PDF]: www-users.math.umn.edu/~olver/ln_/cc.… (ht @academath)
Peter Olver’s home page is chock full of links to his books and lecture notes on variety of topics in mathematics. Lots of pedagogical treasures there. www-users.math.umn.edu/~olver/
To be clear: like many of you, I have doubts about whether Olver’s topological approach to calculus would be “more understandable” for beginners. (My own instinct is to teach calculus very intuitively at first, and hold off on rigor until a 2nd course.) Still, interesting!
For a completely different approach, have a look at “Intuitive infinitesimal calculus” by @viktorblasjo —> intellectualmathematics.com/calculus/. I have not read it, but knowing him, I’m sure I would love it. And check out his manifesto on teaching mathematics: intellectualmathematics.com/manifesto/
