Real analysis has no applications?
As it happens, I just finished teaching a quarter of undergraduate real analysis. I am inclined to rephrase Pugh's statement into a form that I would agree with. If you view analysis broadly as both the theorems of analysis and methods of calculation (calculus), then obviously it has a ton of applications. However, I much prefer to teach undergraduate real analysis as pure mathematics, more particularly as an introduction to rigorous mathematics and proofs. This is partly as a corrective (or at least a complement) to the mostly applied and algorithmic interpretation of calculus that most American students see first.
Some mathematicians think, and I've often been tempted to think, that it's a bad thing to do analysis twice, first as algorithmic and applied calculus and second as rigorous analysis. It can seem wrong not to have the rigor up-front. Now that I have seen what BC Calculus is like in a high school, I no longer think that it is a bad thing. Obviously I still think that the pure interpretation is important. On the other hand, both interpretations together is also fine by me. I notice that in France, calculus courses and analysis courses are both called "analyse mathématique". I think that they might separate rigorous and non-rigorous calculus a bit less than in the US, and it could be partly because of the name.
In fact, it took me a long time to realize how certain non-rigorous explanations guide good rigorous analysis. For instance, the easy way to derive the Jacobian factor in a multivariate integral is to "draw" an infinitesimal parallelepiped and find its volume. That's not rigorous by itself, but it is related to an important rigorous construction, the exterior algebra of differential forms.
Finally, I agree that Pugh's book is great. As the saying goes, you shouldn't judge it by its cover. :-)
I think out of all the fields of mathematics, analysis has the MOST application. We are talking about the subject Newton created to be able to even talk about physics here!
Reading the whole paragraph as reported above, it is clear that it is quite different from the title of this question. Saying that "X-theory involves no application in engineering" just means that an X-theorist, in her job, doesn't employ engineer's tools or language, and as an X-theorist she may even forget about engineering. It definitely does not mean that (i) problems of X-theory were not originated from concrete problems of engineering, nor that (ii) the results of X-theory have no applications in engineering. Actually, at the origin of even the most abstract mathematic theory there are concrete problems of applied science (maybe after several successive steps of abstractions), and also, the final applications are again back in concrete problems.
Abstraction (from abstractus: p.p. of abs+trahere: to take (something) away from) is the usual process by whom we take the essence of a problem in order to focus on it, not to distract ourselves (dis-trahere, to take here and there) by other unessential feature of it, and with the advantage to solve once for all several essentially similar problems.