Spivak's Calculus starts with a set of 13 axioms which characterize the real numbers and then derives all the results you're familiar with in calculus. It's rigorous in the mathematical sense, so if you've never worked through a rigorous math textbook before then this might be a good start since you're familiar with the underlying material.
Here are some exercises to give you a sense of the flavor. If you find these exercises trivial then the textbook might not be for you. If you find them hard, well, welcome to math! :)
These are all before we get to any "calculus." Here "function" means a function of the real numbers.
1. Let f be a function that satisfies the conclusions of the Intermediate Value Theorem. Prove that if f takes on each value only once then f is continuous. Generalize this to the case where f takes on each value only finitely many times.
2. Prove that if n is even, then there is no continuous function f which takes on every value exactly n times.
3. A set A of real numbers is said to be sense if every open interval contains a point of A. Prove that if f is continuous and f(x) = 0 for all numbers x in a dense set A then f(x) = 0 for all x.
4. Find a function which is continuous at every irrational point and discontinuous at every rational point (and prove it as such)
Spivak's Calculus is used as a first-year calculus textbook at lots of schools, so if you find the above even a little challenging or strange-seeming then I'd recommend going through the book.
The last chapter of the textbook is a rigorous construction of the real numbers from the rationals using Dedekind cuts (referenced in the first link).
> Spivak's Calculus is used as a first-year calculus textbook at lots of schools
Umm... where? Not at Stanford, where we used a mainstream, much easier book. So does Princeton. Harvard is famous for having developed a "touchy-feely" calculus book.
Perhaps abroad? It is typical of calculus courses in the US that the students come with fairly weak backgrounds, and a major purpose is to expose and patch holes in the students' backgrounds in algebra and trigonometry.
If you took me to mean the "default" first-year calculus textbook then yes, that's not common. But it's definitely aimed at first-year college students, or at least people who haven't had prior exposure to rigorous mathematical thinking. Compare the style to, say, Spivak's Calculus on Manifolds to see what I mean.
(Edit: I just read your HN bio and know you know the stylistic differences, etc. Sorry!)
Spivak is the first-year Honors Calculus textbook at my alma mater, the University of Chicago. Harvard is also famous for having the most difficult first-year math classes that use even more advanced textbooks like Rudin's Principles of Mathematical Analysis.
My HS background in mathematics was definitely "weak," too. My senior year was the first year my school district ever offered calculus of any stripe in its entire history and I still managed to handle Spivak my first year of college. I took the AP Calculus test on my own and got a 4/5. It's not that crazy.
Here are some exercises to give you a sense of the flavor. If you find these exercises trivial then the textbook might not be for you. If you find them hard, well, welcome to math! :)
These are all before we get to any "calculus." Here "function" means a function of the real numbers.
1. Let f be a function that satisfies the conclusions of the Intermediate Value Theorem. Prove that if f takes on each value only once then f is continuous. Generalize this to the case where f takes on each value only finitely many times.
2. Prove that if n is even, then there is no continuous function f which takes on every value exactly n times.
3. A set A of real numbers is said to be sense if every open interval contains a point of A. Prove that if f is continuous and f(x) = 0 for all numbers x in a dense set A then f(x) = 0 for all x.
4. Find a function which is continuous at every irrational point and discontinuous at every rational point (and prove it as such)
Spivak's Calculus is used as a first-year calculus textbook at lots of schools, so if you find the above even a little challenging or strange-seeming then I'd recommend going through the book.
The last chapter of the textbook is a rigorous construction of the real numbers from the rationals using Dedekind cuts (referenced in the first link).