Are older mathematics textbooks still "valid"? Being interested in learning rigorous calculus (as opposed to the content taught in AP Calculus and intro calculus courses in university), some textbooks mentioned quite often on the internet include Introduction to Calculus and Analysis, Vol. 1 by Richard Courant, Calculus, Vol. 1 by Tom Apostol, and A Course of Pure Mathematics by GH Hardy. These books are regarded as classics. 
My concern, however, is that they are rather old texts, particularly the one by GH Hardy. I do not want to learn from or purchase mathematics books which are inaccurate lest I develop any sort of false beliefs or misconceptions. My question is therefore as follows: are these books, and all relatively older (>5 decades) textbooks in general, sufficiently rigorous by the modern standards of pure mathematics?
 A: Yes
Sometimes old textbooks use rather archaic language and slightly different terminology (like 1915 sort of age) but there is nothing "invalid" (except for maybe errors!) about them.
If it (correctly) proves a claim, that is that. Job done. 
(I love the fact I can buy old editions for a penny (without delivery) on Amazon, no other subject has this! They're always changing!)
A: Broadly speaking, a book's age does not serve to credit or discredit it. That being said, it's a bit like asking, "Is a Magnavox Odyssey still valid"? If it still plays, then it still plays, but you also have to deal with the fact that old things are meant to do different things than new things, and that even if two things have the same goal, time will still help to refine that thing through innovation.
Will Hardy's analysis be correct? Most likely. But you run into two things, especially when you're talking early twentieth century books. For one, the language will be very unorthodox by today's standards. If you're going particularly early in this period then you might even find dissent among authors as to what to actually call what we would refer to now as a "set" (I believe Russell had used "manifold" at some point for what is now a set, and now manifold has a very particular meaning in geometry). Be prepared to Google what words mean, and be prepared for that Googling to be a nontrivial endeavor. It's also important to note that an important part of mathematics is knowing how to read and communicate it, neither of which will be possible if your lexicon is a century old.
Secondly, their methods and approaches will probably not be what we'd use today. This is the refinement. Over time, mathematicians will look for better ways to do what the older folks did. It's common that a classic theorem's proof when first presented will consist of multiple lemmas, and will be a long and drawn out labor, while later authors will "streamline" and "refine" those methods. So though what folks like Hardy might give you is in all likelihood correct, it's also likely that later authors will have improved upon what the classics present. And when I say "improvement", I don't mean that some stuffy journal has found some generalization of a theorem to the point that it's almost unrecognizable; I mean that your typical undergrad text will do in six lines what an older author did in a page and a half, and probably in a reasonable generalization.
EDIT: This newfound brevity might also sometimes be a mater of saying something very similar to what the older authors were saying, but simply having more precise language to say it with.
A: They are rigorous. The issue is with the language. They sometimes use very outdated terms that nobody really uses anymore. Take for example, "Number Theory", by Borevich and Shafarevich. It is a wonderful book, but I really do not know anyone who talks in the language of divisors (from that book). So the problem is that you are not learning anything wrong, but the problem may be that you are learning things in a way different from everyone else talks. 
A: This is mathematical physics rather than maths, but I think as an extreme example is may be helpful.

Asked in 1919 whether it was true that only three people in the world
  understood the theory of general relativity, [Eddington] allegedly
  replied: 'Who's the third?

[Let's disregard the fact that this quote (apocryphal or otherwise) says more about Eddington than theory itself.]
In 1919, four years had passed since publication of GR, so surely many could have accessed and understood it?
Compare today where a rather large number of people would consider that they understand general relativity. There are many standard texts on the subject and it is accessible to undergraduates.
What has happened over the last 100 years is that the articulation of the theory has vastly improved - we have presentations which are much clearer and easier to understand. This progress includes input from a large number of physicists and mathematicians in steadily refining the approach taken. This does not imply in any way that understanding or exposition was in any way "wrong" in 1919.
A: Forgive me if a wax on a little bit on this one, but . . .
I love old math books; and I think, provided one takes the requisite care to "translate" what is sometimes older language, they are a great source of knowledge.  And I think that sometimes being closer in time to "the source" lends a certain clarity of vision.  There's nothing like reading ideas from the guy who first had them, or from his immediate students.
As far as rigor is concerned, well, some books have it and some books don't, and that has little to do with age.  If you can discern a rigorous argument you will recognize one when you see it, no matter how long ago it was committed to paper.  Think Euclid, think Archimedes.
A few of my favorites:  Poincare's New Methods (not sure of the date);  Von Neumann's Mathematical Foundations of Quantum Mechanics (1932); Artin's little book Galois Theory (in the nineteen forties, as I recall);  Courant and Hilbert's Methods of Mathematical Physics (German edition, 1937); another oldie but goodie is Paul Garabedian's Partial Differential Equations (ca. 1965?), old but very clear and rigorous (a grad student of L. Craig Evans (yes, the Evans) told me it was outdated); all are "antiquated" by the standards of 2015, but what great sources!
Not to leave physics out altogether, will Feynman's Lectures ever be outdated?  How about Dirac's Principles of Quantum Mechanics?  True, these books adhere to a physicist's standard of rigor, but there is a lot of math to be understood from both of these guys; so do I opine.
An anecdote to close:  when I was a kid learning calculus, I struggled and struggled with Thomas, a de facto standard in the 1960s.  Then my old maid aunt--she was a math teacher on a Navajo reservation--gave me her copy of Lynman Kells' Calculus (1949).  The scales then fell from my eyes . . . 
A: There are beautifully written books out there - some of them are really old (e.g. Euclid's elements) and some of them are new. The same is true for badly/careless written books - in mathematics and elsewhere.
Therefore this question cannot have a definitive answer. However, there is more to an enjoyable textbook than rigor and amongst them is terminology, which tends to change drastically over the decades. I've never tried but my advisor mentioned on more than one occasion how painful to read Gödel's original papers are - by today's standards.
Another example is Cohen's 1963 paper, in which he invented forcing. Shoenfield and Solovay (as far as I know) dramatically eased this method and made it thereby both more accessible and enjoyable.
Depending on the subject, the flavor of a textbook may also change heavily over the years to an extend, that two books consider completely separate objects to answer basically the same questions. Additionally, an older book may lack more recent result - either because they weren't in reach at that time, or because they simply weren't known.
These are only some of the aspects to be considered, when choosing a textbook on a given subject...
A: I studied analysis 40 years ago, I don't think there's anything new that would be presented as undergrad material, but perhaps the style and nomenclature has been 'modernized', so you might find the classics a bit harder to follow, but I don't think you'll find errors or inaccuracies in the proofs - these texts have been around for long time and have been reprinted and revised over the years.  I used Apostol for undergrad, I always liked his proofs, generally very elegant and tight. 
I can't seem to find my Hardy Pure Mathematics - as my memory goes, it was a great reference, all the usual stuff you would need for calculus and analysis was there, but was written before textbooks used sets so heavily, so you'll see a bit of a difference between Hardy and Apostol, Hardy is a bit more intuitive whereas Apostol is more formal ( modern). 
You would benefit from an appreciation of what a good proof takes, even if this is beyond the scope of the AP level mathematics. 
regards.
