Book that presents analysis using minimal premises Can someone recommend me an analysis book in which every (or at least most) theorem(s) are given using minimal premises ?
By this I mean that most analysis books in which I looked present give the theorems under premises that are a little bit to strong, for example they often require $f$ to be continuous where it would suffice for $f$ to be $R$-integrable or (in the case of second order derivatives) they require that $f$ is continuously differentiable in an interval around $a$ (if $a$ is the point in which the second order derivative is to be taken), where it would suffice to merely state that $a$ is an accumulation point for the "second order difference function" which in the limit to $a$ gives the second order derivative.
 A: I think that rather than looking for such a book, you should have a lot of analysis books on hand (purchase, borrow from library, etc.). Then, when you come to a certain topic in your primary book/text, you can look up that topic's treatment in other books. You'll often find that Author A gives a more refined version than Author B of Theorem X, but then Author B gives a more refined version than Author A of Theorem Y. Also, the push towards having the least hypotheses can quickly lead to technical issues beyond the scope of your current interests. For example, if you really want something that involves fairly minimal hypotheses, take a look at Krishna M. Garg's book Theory of Differentiation: A Unified Theory of Differentiation Via New Derivate Theorems and New Derivatives. Here, instead of a hypothesis of continuity or even of semicontinuity on an interval, you'll find a hypothesis that may be something like "unilaterally quasi-semicontinuous on a co-countable set". However, my guess is that Garg's book is not what you're looking for. That said, I suggest looking at Andrew Gleason's Fundamentals of Abstract Analysis.
