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.