Part I

One way and another, I got a good background in real analysis. So, okay, I'll try to answer:

An answer depends on what is meant by real analysis.

Part of the answer is advanced calculus, and part of that is the Gauss, Green, and Stokes theorems. If do these the modern and high end ways, then get as deep as like, and spend as much time as like, in differential geometry, calculus on manifolds, differential forms, E. Cartan, exterior algebra, algebraic topology, etc. But for 19th century physics and engineering, there is a way to get a good treatment of what need in about a nice weekend from (own TeX markup):

Tom M.\ Apostol, {\it Mathematical Analysis: A Modern Approach to Advanced Calculus,\/} Addison-Wesley, Reading, Massachusetts, 1957.\ \

Note how old it is! It's no longer really "modern"! Get a used copy; that's what I did. You don't want a newer edition!

But with that out of the way, might take a fast pass through an old MIT standard

Francis B.\ Hildebrand, {\it Advanced Calculus for Applications,\/} Prentice-Hall, Englewood Cliffs, NJ, 1962.\ \

There's a lot of fun stuff in there, but it's TOO short on proofs. So, I'd pass it up and return later when know enough real analysis to guess or understand most of the proofs easily.

Note: In (the relatively elegant and easy to read)

George F.\ Simmons, {\it Introduction to Topology and Modern Analysis,\/} McGraw Hill, New York, 1963.\ \

the flat statement IIRC "The two pillars of analysis are linearity and continuity."

Well, for the linearity, really need a good background in linear algebra. For this, you need at least three books, a really easy one, elementary, that starts with, say, systems of linear equations and Gauss elimination. Then a more advanced one that emphasizes the axioms for a vector space, does vector spaces with at least both of the real and complex numbers (over finite fields can be important for computing but not for real analysis) and emphasizes eigenvalues and eigenvectors, and finally the grand one, written at the knee of von Neumann as a finite dimensional introduction to Hilbert space theory,

Paul R.\ Halmos, {\it Finite-Dimensional Vector Spaces, Second Edition,\/} D.\ Van Nostrand Company, Inc., Princeton, New Jersey, 1958.\ \

If you get very far in real analysis, then you will want a good treatment of at least basic Hilbert space theory, and Halmos is a good start.

For his section on multi-linear algebra, I'd skip that unless plan to take the exterior algebra of differential forms seriously. In that Halmos book, concentrate on vector spaces, vector subspaces, linear transformations, eigenvalues, eigenvectors, and Hermitian and unitary matrices. Also at the end note the cute ergodic theorem! The big deal in about the last half of the book is spectral decomposition -- don't skip that.

For your second book, I used E. Nearing -- he was a student of E. Artin at Princeton. So, Nearing's book is high quality stuff. I worked carefully through that and learned a lot. But his appendix on linear programming is a disaster! Can do nearly everything important in linear programming and its simplex algorithm as just a simple -- learn it in an hour -- extension of Gauss elimination. Nearing does finite cones and dual cones for which he never makes a clear connection with linear programming. And although he works with all those cones, still he misses the theorems of the alternative -- Farkas, etc. -- important in parts of optimization, convexity, etc.

Also recommended is Hoffman and Kunze and, IIRC, available for free on the Internet as a PDF file.

There is much more in linear algebra, e.g., from R. Bellman, R. Horn, on numerical methods, etc. but these are not crucial for a rush to real analysis.

Maybe part of real analysis, that is, advanced enough, is "Baby Rudin", Principles of Mathematical Analysis. The later editions have near the end some tacked on material, mostly without sufficient context, on the exterior algebra of differential forms. Skip that. If you want that material, then go for R. Buck, Advanced Calculus or Spivak, Calculus on Manifolds or really just go for a real book on differential geometry, manifolds, calculus on manifolds, etc. Such differential geometry is from important to crucial for several objectives but is NOT on the mainline of a rush to real analysis.

So, what is going on in Baby Rudin? Okay, the main idea of the book is that we can give a solid development of the Riemann integral for a function that is continuous on a compact set. So, get to learn about continuity, that is, one of the two pillars of analysis. Then hand in hand with continuity is compactness, so get to see that. All of that is in just the first few chapters; that's what those first few chapters are all about -- continuity and compactness. E.g., get to learn that in R^n (for the set of real numbers R and a positive integer n), a set is compact if an only it is closed and bounded -- super, important, crucial stuff, the key to a clean up of Riemann integration, that is, material Newton didn't know.

Then with that material on continuity and compactness, Rudin does the Riemann–Stieltjes integral. There, mostly just ignore the Stieltjes part with its possibility of step functions (maybe as a cheap answer to what the physics people try to do with the Dirac delta function, which, of course, is not really a function, but has a clean fix-up with distributions and measure theory) and just read that Rudin material for the Riemann integral of first calculus. You will likely never see the Stieltjes extension again.

The main idea is: A function continuous on a compact set is also, presto, bingo, wonder of wonders, really nice day, uniformly continuous, and that makes the derivation of the Riemann integral really easy. Really, that's the core idea of the whole book. Baby Rudin can seem severe, but with this introduction you should be able actually to like it a lot. Later in the book, Rudin touches on the fact that the uniform limit of a sequence of continuous functions is continuous -- same song, next verse. It was a Ph.D. qualifying exam question for me; I did get it.

Later he gives a really nice treatment of Fourier series -- that is very much worth reading.

I'd suggest one side trip: Cover the inverse and implicit function theorems. They are just a local, non-linear generalization of what you will see really easily for linear transformations in linear algebra via, right, just Gauss elimination. For a source? There is a good treatment in W. Fleming, Functions of Several Variables. IIRC, there is also a cute proof based on contractive mappings.

So, by then you will have a good start on both continuity and linearity.