Reddit mentions of Introductory Real Analysis (Dover Books on Mathematics)

The answer is "virtually all of mathematics." :D

Although lots of math degrees are fairly linear, calculus is really the first big branch point for your learning. Broadly speaking, the three main pillars of contemporary mathematics are:

1. Analysis
   
2. Algebra
   
3. Topology
   
   You might also think of these as the three main "mathematical mindsets" — mathematicians often talk about "thinking like an algebraist" and so on.
   
   Calculus is the first tiny sliver of analysis and Spivak's Calculus is IMO the best introduction to calculus-as-analysis out there. If you thought Spivak's textbook was amazing, well, that's bread-n-butter analysis. I always thought of Spivak as "one-dimensional analysis" rather than calculus.
   
   Spivak also introduces a bit of algebra, BTW. The first few chapters are really about abstract algebra and you might notice they feel very different from the latter chapters, especially after he introduces the least-upper-bound property. Spivak's "properties of numbers" (P1-P9) are actually the 9 axioms which define an algebraic object called a field. So if you thought those first few chapters were a lot of fun, well, that's algebra!
   
   There isn't that much topology in Spivak, although I'm sure he hides some topology exercises throughout the book. Topology is sometimes called the study of "shape" and is where our most general notions of "continuous function" and "open set" live.
   
   Here are my recommendations.
   
   Analysis If you want to keep learning analysis, check out Introductory Real Analysis by Kolmogorov & Fomin, Principles of Mathematical Analysis by Rudin, and/or Advanced Calculus of Several Variables by Edwards.
   
   Algebra If you want to check out abstract algebra, check out Dummit & Foote's Abstract Algebra and/or Pinter's A Book of Abstract Algebra.
   
   Topology There's really only one thing to recommend here and that's Topology by Munkres.
   
   If you're a high-school student who has read through Spivak in your own, you should be fine with any of these books. These are exactly the books you'd get in a more advanced undergraduate mathematics degree.
   
   I might also check out the Chicago undergraduate mathematics bibliography, which contains all my recommendations above and more. I disagree with their elementary/intermediate/advanced categorization in many cases, e.g., Rudin's Principles of Mathematical Analysis is categorized as "elementary" but it's only "elementary" if your idea of doing math is pursuing a PhD. Baby Rudin (as it's called) is to first-year graduate analysis as Spivak is to first-year undergraduate calculus — Rudin says as much right in the introduction.

You should check out an Infinitely Large Napkin. It’s a good book for building up intuition. Unfortunately the probability section isn’t complete yet, but I think you would enjoy the measure theory section.

If you want some more formal books, there’s Kolmogorov’s Introductory Real Analysis and Folland’s book on Real Analysis

Many of the current Dover books were used as textbooks not too long ago when they were still in print from the original publisher. For example, this was the book used in my undergrad intro to PDEs course, and this was used for the undergrad intro to algebraic topology. Those were the hardcover versions, before Dover bought them. Even the Kolmogorov & Fomin book was used in grad school for real analysis.

To me the real Dover gems are the old ones (pre-1960) that are hard to find now except at used book stores or on Amazon. Though they may seem "outdated" many of them actually cover topics that are still useful but not covered nowadays.

If you want to learn serious mathematics, start with a theoretical approach to calculus, then go into some analysis. Introductory Real Analysis by Kolmogorov is pretty good.

As far as how to think about these things, group theory is a strong start. "The real numbers are the unique linearly-ordered field with least upper bound property." Once you understand that sentence and can explain it in the context of group theory and the order topology, then you are in a good place to think about infinity, limits, etc.

Edit: For calc, Spivak is one of the textbooks I have heard is more common, but I have never used it so I can't comment on it. I've heard good things, though.

A harder analysis book for self-study would be Principles of Mathematical Analysis by Rudin. He is very terse in his proofs, so they can be hard to get through.

>what is the difference really between 'calculus' and 'real analysis'

At the undergraduate level, "calculus" typically means the what. For example: what is this limit? What is the derivative of a given function? What is the value of this integral?

"Analysis" more typically gets into the why behind calculus. Why does this function have a limit? Justify why the typical rules for differentiation—product rule, chain rule, etc.—are valid. Define what it means for a function to be integrable over a given interval, and justify your computation of a given integral.

There's a lot more going on than just that, but to first approximation, making the distinction between the what of calculus and the why of analysis is a good starting point.

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I don't have a copy of Kolmogorov's text, so I'm at a disadvantage. I assume you mean something like this book in the Dover series? If so, then the table of contents suggests it's a pretty ambitious book, at least for typical undergraduates—and especially if it's one's introduction to the subject matter. That text by Kolmogorov covers some of both metric space topology and point-set topology, as well as linear algebra, measure theory, integration, and differentiation (itself in the context of Lebesgue integration). I'm no expert on the matter, but Kolmogorov's (and Fomin's) text seems more representative of what's often called "functional analysis" rather than just "real analysis". I suspect that pedagogically, you might benefit from a more "concrete" introduction to real analysis before tackling something like this textbook.

As for the inverse and implicit function theorems, there are a handful of ways to approach those results. One way is to show that the two theorems are equivalent: the inverse function theorem is true if and only if the implicit function theorem is true. The way a lot of books proceed is to establish the inverse function theorem by making some suitable simplifications—e.g., that the derivative map is being evaluated at the origin, and that this derivative map is the identity map—then apply the contraction mapping theorem. (Of course, the two theorems are equivalent, so one could instead prove the implicit function theorem first, instead.)

Rudin is emphatically not the only suitable textbook for something like this, but nearly any such "suitable" textbook will inevitably be challenging. It will help you considerably to have already had linear algebra, at least, especially if you turn to a textbook that presupposes linear algebra as a prerequisite. I'm not sure what to recommend to you, but here are a few textbooks I've used over the years (in addition to those already mentioned above):

• Elementary Classical Analysis, Marsden and Hoffman
  
  
• Mathematical Analysis: An Introduction, Browder
  
  
• Introduction to Classical Real Analysis, Stromberg
  
  
• Advanced Calculus of Several Variables, Edwards
  
  Stromberg and Browder are challenging, on the general level of Rudin in that respect. Marsden and Hoffman has an unusual structure (at least in my volume), where the statements of the theorems and propositions are separated from the proofs of those assertions. (And, if memory serves, M&H may have suffered from a number of typos.) Edwards may be the most accessible of the books above, and it covers quite a bit from both a "classical" and higher-order "theoretical" perspective. But which of these books—if any—would be the best fit for you would necessarily be a matter of speculation for me.
  
  I hope this was nonetheless helpful. Good luck!

I probably should have waited for a response but I bought. https://www.amazon.co.uk/Introductory-Analysis-Dover-Books-Mathematics/dp/0486612260 yesterday as it's so much cheaper. So hey let's hope it's worth the £5 I paid

The only serious next step.