Walter Rudin’s classic book, often called “baby Rudin”, is one of the most famous mathematics books. The third and final edition was published in 1976 and it remains both loved and hated by many mathematicians.
It’s somewhat infamous for its conciseness and for being both tough to read and for having difficult exercises. This is true to a large extent and it’s also what makes the book charming and interesting. It’s not the easiest way to get to know mathematical analysis, but it’s a great way to improve your mathematical abilities and to get plenty of exposure to a masterfully compact presentation of the topic where all the definitions and theorems fit together in a well-laid puzzle! The book presents a rigorous mathematical theory of basic mathematical analysis, including a construction of the real numbers. The fact that the book is as famous and well-read as it is also means that it presents the subject in a way that is already familiar to many mathematicians across the globe.
In the end, it’s not hard to understand given some exposure to the subject before. As with any mathematics book, it does take time to work through, however. There are a few small misprints (at least in my international edition) early in the book, but it only serves to keep the reader alert.
In short, the book remains a classic and well worth reading. There are other good books that are also excellent choices, such as Real Mathematical Analysis by Charles Pugh, but Rudin’s style and the abstract and theoretical presentation of “baby Rudin” is very pleasing. My recommendation is to read another book to learn mathematical analysis and then read Rudin to get familiar with a great classic and to get one more encounter with the material. Then read other books on more specialized topics.
At least for me, the book did explore some theorems and topics that I hadn’t encountered much before. There are plenty of little gems in there. For example, I really enjoy the theorem that shows how the rational numbers are dense among the real numbers despite there being more real numbers.
Should you solve all the exercises? Well, the typical recommendation would be to do so. Personally, I didn’t. The exercises are great and theoretical, but I was mainly interested in getting familiar with the book’s presentation of the material right now. I had a look at the exercises and solved some, which I highly recommend even if you share my goal.
The whole book is good, but I agree with others that the first 8 chapters are the best ones. It would make perfect sense to only read these or skim the rest.
I recommend the book for anyone who is interested in pure mathematics and want to read a true classic in the subject, especially if it’s not your first. There’s a good review from MAA that goes into more detail.