Which should a high schooler complete first? Hardy's "A Course of Pure Mathematics", or Ghorpade & Limaye's "A Course in Calculus and Real Analysis"? I am a high schooler. I am looking forward to go through books that introduce calculus and analysis simultaneously. I have learnt to write proofs and have studied propositional calculus. I have chosen two texts to read, i.e, Hardy's A Course of Pure Mathematics, and A Course in Calculus and Real Analysis by Sudhir Ghorpade and Balmohan Limaye, Springer.
Which one should I read?
(I am a complete beginner in calculus. So, I doubt Hardy is the one for me. But some recommend it for beginners.)
 A: The second book you suggest, A Course in Calculus and Real Analysis, is maximal in the set of Calculus books partially ordered by quality.
In other words there is no better book, although there might be excellent books which cannot be said to be better or worse than this one.
Its main qualities are:

*

*The incredible quality of the colour illustrations which almost evoke an art book.
For example, look at the weird piecewise linear zigzag function on page 27.


*The Notes and Comments at the end of each chapter, which really explain what is going on under the hood and give extremely interesting information on the history of the topic discussed in the chapter.


*Each of these topics is discussed in depth and many results are quite interesting and unusual in a textbook, as for example on page 78 the discontinuous function having the intermediate value property.
Another result (not trivial at all despite appearances) that I have never seen discussed in a Calculus textbook is that the logarithm is a transcendental function: the proof is found on pages 270-271.
And as an algebraic geometer I was astounded and delighted to find on pages 143-144 a discussion of the multiplicity of a plane algebraic curve at one of its points: I am not aware of any Calculus book having ever discussed that subject!


*The many exercises (around 500), some quite challenging and/or instructive.
For example in the exercises to Chapter 9 you can learn about Raabe's test and the hypergeometric series.
And what about the other book, A course of Pure Mathematics ?
It was written in 1908 by Hardy, the best 20th century British analyst, and helped revive Analysis in Great Britain, a country that lagged far behind the continental school in that subject.
Britain was brilliant in Algebra, Geometry and above all Applied Mathematics, but was no match for the illustrious analysts across the Channel: Weierstrass, Riemann, Volterra, Hermite, Cauchy, Jordan, de la Vallée-Poussin and dozens of others.
Hardy wanted to open his compatriots to modern Analysis and a stepping stone was the book under discussion, written in the style of   "a missionary preaching to cannibals", as he himself wittily acknowledges in the preface to the seventh edition.
I love that book for its beauty and depth, but also for nostalgia's sake.
To sum up Hardy's is a great book to browse and enjoy, but for a young mathematician wanting to learn the subject pleasantly but in depth I  warmly recommend A Course in Calculus and Real Analysis.
