What should I study real analysis or calculus?

Question

So I can do elementary single variable calculus (except series), also I am familiar with $\varepsilon$-$\delta$ definition of the limit, and also read the definition of Darboux integral, but don't remember it fully. I want to self-study more by myself. And I don't know whether I should choose some not so rigorous calculus book (which maybe includes some multivariable calculus) but which will teach to do examples or some rigorous real analysis book. So what would you suggest me to choose and what book would you suggest?
Thanks in advance.

Answer 1

I am italian, and the usual program in our university starts more or less directly with Mathematical Analysis. However, in high school we learn basic (really basic) calculus rules.

According to your description, I would suggest to review your knowledge of calculus, and to get acquainted with advanced calculus (functions of several real variables). You should move on to Mathematical Analysis once you know by heart all the besic definitions of calculus (limits, continuity, derivative, integral).

I like Serge Lang's books (published by Springer-Verlag), but the celebrated books by Tom Apostol (three volumes on calculus) could be a good choice. If you feel ready to learn some hard mathematical analysis, with rigorous proofs of everything, you can read Walter Rudin's Principles of mathematical analysis.