Publisher's Synopsis
A course dealing with the fundamental theorems of infinitesimal calculus in a rigorousmanner is now recognized as an essential part of the training of a mathematician. Itappears in the curriculum of nearly every university, and is taken by students as "AdvancedCalculus" in their last collegiate year, or as part of "Theory of Functions" in the firstyear of graduate work. This little volume is designed as a convenient reference book forsuch courses; the examples which may be considered necessary being supplied from othersources. The book may also be used as a basis for a rather short theoretical course on realfunctions, such as is now given from time to time in some of our universities.The general aim has been to obtain rigor of logic with a minimum of elaborate machinery.It is hoped that the systematic use of the Heine-Borel theorem has helped materiallytoward this end, since by means of this theorem it is possible to avoid almost entirelythe sequential division or "pinching" process so common in discussions of this kind. Thedefinition of a limit by means of the notion "value approached" has simplified the proofsof theorems, such as those giving necessary and sufficient conditions for the existence oflimits, and in general has largely decreased the number of "'s and 's. The theory of limitsis developed for multiple-valued functions, which gives certain advantages in the treatmentof the definite integral.
