This little book is especially concerned with those portions of advanced calculus in which the subtlety of the concepts and methods makes rigor difficult to attain at an elementary level. The approach taken here uses elementary versions of modern methods found in sophisticated mathematics. The formal prerequisites include only a term of linear algebra, a nodding acquaintance with the notation of set theory, and a respectable first-year calculus course one which at least mentions the least upper bound sup and greatest lower bound inf of a set of real numbers. Beyond this a certain perhaps latent rapport with abstract mathematics will be found almost essential.