> (I'm a bit biased here since I'm really having a hard time with baby Rudin)
Here is some help with Baby Rudin:
He wants to get to the Riemann integral, that is, the simplest version of ordinary integration in first calculus. Then for a little more, he also wants to do the Stieltjes extension of the Riemann integral.
So, he wants to integrate a function of one real variable -- he wants to keep it simple and elementary.
For this he wants to state carefully the properties of the function he wants to assume and use. Again, he goes for the simple stuff:
First he integrates the function f only over an interval, e.g., [0,1] or in general [a,b] for real numbers a and b with a < b (maybe a <= b, I save the effort of checking my copy). So, the interval is of finite length, b - a. Integrating over the whole real line from minus infinity to infinity is more difficult and left for later (and is much better done with the Lebesgue integral, see below). Also the interval is closed, that is, includes the two end points a and b.
Second, he wants the function f to be continuous on that closed interval.
So, he needs to define continuous. And what is magic about the closed interval [a,b] is that it is compact. So, he wants to define compact.
Then with continuous and compact, he shows that the function is not just continuous but also uniformly continuous, and that is the crucial, central, key property of function f that makes defining the Riemann integral easy, showing that the integral exists, and establishing its basic properties.
So, Rudin takes you off on a little side journey to understand closed, continuous, compact, and uniformly continuous. Also, with the real numbers, he uses the completeness property -- the rationals are not complete and won't work; he needs the reals! It's good to know this. Completeness also generalizes, e.g., is a key property in Hilbert space and Banach space.
Now he wants to give you a little more for your money, effort, time, etc.: These concepts of completeness, closed, continuous, compact, and uniformly continuous generalize, that is, are principles as in the title of his book. So, really he is attacking, i.e., defining and developing, the simplest integral of freshman (or high school) calculus with more general concepts, closed, continuous, compact, uniformly continuous, and completeness.
And while his function f has only one real variable, he generalizes a little, takes a positive integer n and the set of real numbers R, and considers the Euclidean n-space R^n. Then he proves that in R^n, a set is compact if and only if it is both closed and bounded. Darned good to know! And the proof is not difficult and worth understanding. I used this result in a paper I published on some really tricky aspects of the Kuhn-Tucker conditions. My department got all impressed, and that eased my path to my Ph.D.
Compact: (A) Every infinite subset has a limit point. (B) Every open cover has a finite subcover. Compact via (A) and (B) is so nice that it is almost as well behaved as only finitely many points and generalizes enormously.
A real valued, continuous function with a compact domain is forced to be quite nicely behaved -- in particular to make the Riemann integral easy to develop.
Then the uniformly continuous says that if we pick a fine partition on the X-axis (the domain of function f), then the resulting partition we get on the Y-axis (the range of function f) is also fine -- so, as we make the partition fine on the X-axis, we can be sure the partition on the Y-axis will be as fine as we please which means that, with completeness, the finite Riemann sums converge and our Riemann integral is defined.
So, completeness, closed, bounded, compact, continuous, uniformly continuous -- not very difficult and worth understanding.
Then he moves on and does the basics of infinite sequences and series, enough to define logs, exponents, and the trig functions.
Rudin is especially good with Fourier theory, and there in Baby Rudin gives a solid treatment of Fourier series. Good to know. Of course, first see the connections with organ, flute, and violin music.
Last I checked he did do the inverse and implicit function theorems -- central in Lagrange multipliers and differential geometry.
And he showed that more generally the Riemann integral exists if and only if the function is continuous everywhere except on a set of measure zero -- so he touches on measure theory.
Somewhere you should see a good proof of Leibniz's Rule (differentiation under the integral sign).
Now that you see the Riemann integral, Lebesgue's approach is better: Lebesgue partitions on the Y-axis. Turns out, for handling pathological cases, that works out a lot better. It also generalizes nicely and, in particular, is a great foundation for probability theory. That is, expectation in probability is just a Lebesgue integral. Very nice.
Rudin's development of the Lebesgue integral in his Real and Complex Analysis is very nice. Royden's treatment is a little easier to read. Read them both. And there Rudin also has very nice chapters on the Fourier transform, Banach space, and Hilbert space. Read those three chapters and apparently be for all your career nicely ahead of nearly everyone in physics, chemistry, and engineering. E.g., see how the Heisenberg uncertainty principle, whatever it is in the natural world, is really just a simple result in Fourier theory.