I didn't see any suggested books from the great Russian school of mathematics, here is a brief list of superb, well written, example oriented books:
- Elements of the Theory of Functions and Functional Analysis by A. N. Kolmogorov and S. V. Fomin
- Theory of Functions of a Real Variable by I. P. Natanson (this I think, it's hard to find)
- Theory of Functions of a Complex Variable, Second Edition (3 vol. set) by A. I. Markushevich
- Elements of Functional Analysis by L.A. Lusternik and V.J. Sobolev
- Problems in Mathematical Analysis by B. Demidovich
- Calcul intégral et differentiel (2 vol. set) by N. Piskounov. In English should be something like Differential and Integral Calculus
- A Course of Mathematical Analysis (2 vol. set ) S. Nikols'skii
- Differentialrechnung und Integralrechnung (3 vol. set), by Gregor M. Fichtenholz. Unfortunately, there is no English translation of this book, only the German translation that it's mentioned. I think this was THE calculus book on Russia. THIS BOOK SHOULD BE TRANSLATED INTO ENGLISH, and I suspect that there is no copyright, it appears around 1959 I think.
- Mathematical Analysis (2 vol. set) by Vladimir A. Zorich. This is a very recent book from a great mathematician which is in Moscow university. The book is based upon is lecture.
I have also found a Spanish translation of a book written by S. Banach about "Differential and integral calculus". It is very good as an undergraduate book. The spanish translation (for those who want to search) is Calculo diferencial e integral
Books that undergraduate should not touch (in my humble opinion) are books written in bourbaki's style.
