Real analysis is the mathematics that concerns real numbers and constructions based on real numbers. In many respects, it is a more formal version of calculus. The parts of real analysis that aren't taught in calculus tend not to be directly important for data science. However, real analysis is required for advanced probability and understanding certain kinds of transformations, which are both important for data science. It's also an interesting subject in its own right. You must be familiar with writing mathematical proofs in order to study real analysis.
Recommended Books

Principles of Mathematical Analysis
Walter Rudin
 Intext exercises
 Widespread solutions to exercises online
This is one of the best mathematics books ever written. It is concise, and it is more comprehensive than most introductory books. The drawback of this book is that it is not a particularly gentle introduction to real analysis. However, if you can make it through the challenges of Rudin you'll be wellprepared to learn advanced measure theory based topics in data science. It's also very useful as a reference. If you want discussion rather than exposition, you'll have to look elsewhere, such as at our other recommendation.

Understanding Analysis
Stephen Abbott
 Intext exercises
 Errata
This book is a gentle introduction to real analysis, and it contains discussion that's worth reading even if you're already familiar with real analysis. However, it's not as comprehensive as Rudin, so we recommend proceeding on to Principles of Mathematical Analysis if you decide to go with this book.