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.

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Analysis is a huge subject - which parts are you looking for? What is your background/preliminaries? – AD. Oct 20 '11 at 10:48
Walter Rudin wrote some excellent books in analysis. – AD. Oct 20 '11 at 10:49
@AD. My preliminaries are fairly low, mening I'm looking for a book that treats (rigorously) the analysis of functions of one and several real variables (if its analysis in Banach spaces its also ok, as long as the connections to the analysis of reals variables are made) – resu Oct 22 '11 at 9:07