Reddit reviews Introduction to Classical Real Analysis (Wadsworth & Brooks/Cole Mathematics Series)
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>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):
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!