Understanding the foundations of Calculus? What branch of mathematics should someone study to really understand why calculus works ? I know basic calculus but for me it seems really more like apply the tools to compute stuff. I would like to know why it is true.
Any help ?
 A: As commented, the actual answer is real analysis.
Now, you say you want to know why calculus works. There are proofs in most calculus books, you know. Those proofs do explain why things work.
Finally, Spivak Calculus is an excellent calculus book, with much more emphasis on proofs than usual.
A: Its great that you are interested in foundations of calculus (normally even book authors and teachers are not so interested in teaching foundations of calculus to students who are learning calculus for the first time at an age of 16 years or so).
The teaching of calculus follows almost the same pattern in most countries:


*

*First one is taught calculus as a weird complicated tool which has many many interesting applications within and outside of mathematics. The focus here is on learning techniques and tactics of calculus to apply it for various practical problems. Proofs are almost never provided by using the banal excuse that "proof of this theorem is beyond the scope of the book/syllabus" and some book authors commit intellectual fraud by giving incorrect/non-rigorous/intuitive proofs and students think that those proofs are real.

*Next the student is taught the same concepts of calculus (plus a few more abstract ones) and this time the focus is on proofs and foundations and applications within mathematics. And then it is no longer called calculus but rather real-analysis.


In most cases a student does not have the opportunity to study both the above courses and his/her charm of calculus is limited only to the practical applications of calculus. Missing "the beauty of the simple and elegant theory of real numbers and the edifice of calculus built on top of it" is something very very unfortunate for many students.
Luckily I was fortunate enough to get hold of Hardy's A Course of Pure Mathematics when I was going through a first course in calculus. Do read this book not just for the foundations of calculus, but for the sheer joy of reading mathematics. 
