TITLE: Are older mathematics textbooks still "valid"?
QUESTION [37 upvotes]: 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?

REPLY [4 votes]: 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.