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 for…
Starting there, as you go on in real analysis, you will place much less emphasis on continuity. The main work will replacing the Riemann integral by something that does much better on the edge or pathological cases.
Well, the Riemann integral was, right, a lot about area. Well, it's actually not a really good theory of area -- doesn't have all the nice properties we would wish for. So, that view of area gets polished up and replaced by measure theory. Next, that theory of area is used to define the Lebesgue integral, which is much nicer than the Riemann integral, e.g., gets rid of the assumptions of a compact set and continuity.
Lebesgue was a student of E. Borel and did his work near 1900. You will see the Heine-Borel theorem in Baby Rudin.
In simplest terms, the Lebesgue idea is to partition on the Y axis instead of the X axis like the Riemann integral does. Just why that works so well is cute. It also frees up the domain of the function to be much more general, and in 1933 A. Kolmogorov used that fact, finally, to give a good foundation for probability theory, the one accepted now in essentially all advanced work in probability, stochastic processes, and mathematical statistics.
Likely the nicest book to read on real analysis is H. Royden, Real Analysis. But also, later, read the first, real half of W. Rudin, Real and Complex Analysis.
Note: For real analysis, complex valued functions of a real variable are important; functions of a complex variable are quite different and not important; Rudin has some deep reasons to disagree with me. Rudin is no doubt correct; for students trying to learn, I'm correct!
Once you have the main ideas in mind, and maybe what I've given here will be enough, Rudin is more succinct and nicer to read than Royden. But read them both, Royden first. Go through Royden quickly since will do it all again in Rudin.
In Royden, notice his Littlewood's Three Principles as a cute view of what is going on in measure theory and the Lebesgue integral. Royden also has some exercises on upper and lower semi-continuity -- glance at those and get the main idea for the remaining connection with continuity. Notice the nice treatment of differentiation -- the connection between integration and differentiation is weaker for the Lebesgue integral because the integral is so much more general. Then notice the Carathéodory extension result -- that is the key to defining measures on the real line, in particular, Lebesgue measure.
Spend at least a weekend in
John C.\ Oxtoby, {\it Measure and Category:\ \ A Survey of the Analogies between Topological and Measure Spaces,\/} ISBN 3-540-05349-2, Springer-Verlag, Berlin, 1971.\ \
An amazing weekend.
Keep at hand
Bernard R.\ Gelbaum and John M.\ H.\ Olmsted, {\it Counterexamples in Analysis,\/} Holden-Day, San Francisco, 1964.\ \
Notice that for the Lebesgue integral, there is one, central proof technique in just four steps: Prove the theorem for a function that is just a positive constant defined on a set, say, just a box. By linearity, generalize to finite sums of such functions. Then by monotone continuity (see early on the monotone convergence theorem), prove the result for all non-negative functions. Last by linearity again, get the result for all integrable functions.
Okay, for integrable, there is a cute approach, trick: Define the integral for non-negative functions, even ones defined on the whole real line or all of R^n (for the set of real number R and a positive integer n) or an abstract measure space. Okay, easily enough, that integral can have value positive infinity (the Riemann integral gets sick here). Then do the same for a function that is all For this four step proof technique, can knock off a really nice version of Fubini's theorem, that is, interchange of order of integration.
The Lebesgue integral with the dominated convergence theorem, which will see early on, give nice versions of differentiation under the integral sign.
Note: Commonly in physics, engineering, probability theory, etc. we write integrals over the whole real line. Alas, really, for Riemann integration, the appropriate theorems are rarely presented; they are not in Baby Rudin. Really, the Lebesgue integral is needed. So, you've been needing the Lebesgue integral for a long time.
Then get to apply the Lebesgue integral to the powerful Radon-Nikodym theorem (Rudin gives von Neumann's cute proof based, amazingly, in part on just polynomials), get the main, important duality theorems (the Lebesgue integral is the main linear operator!), Banach spaces, Hilbert spaces, and the Fourier integral.
Cover that and can claim you have a good start on real analysis. You will also have apparently so far the only good background for probability, stochastic processes, and mathematical statistics. E.g., the Radon-Nikodym theorem is crucial for conditional expectation, Markov processes, martingales (amazing things with one of the strongest inequalities in math and a super short proof of the strong law of large numbers), ergodic theory, sufficiency in statistics, a quite general proof of the Neyman-Pearson result in statistical hypothesis testing, and more, e.g., the role of Brownian motion in potential theory, stochastic optimal control, ....
Go for it!