Lectures on the Theory of Functions of Real Variables, Vol. 1 (Classic Reprint) - Tapa dura

Sinopsis

Excerpt from Lectures on the Theory of Functions of Real Variables, Vol. 1

The present work is based on lectures which the author is accustomed to give at Yale University on advanced calculus and the theory of functions of real variables. It falls in two volumes, and the following remarks apply only to the first.

The student of mathematics, on entering the graduate school of American universities, often has no inconsiderable knowledge of the methods and processes of the calculus. He knows how to differentiate and integrate complicated expressions, to evaluate indeterminate forms, to find maxima and minima, to differentiate a definite integral with respect to a parameter, etc. But no em phasis has been placed on the conditions under which these pro cesses are valid. Great is his surprise to learn that they do not always lead to correct results. Numerous simple examples, how ever, readily convince him that such is nevertheless the case.

The problem therefore arises to examine more carefully the conditions under which the theorems and processes of the calculus are correct, and to extend as far as possible or useful the limits of their applicability.

In doing this it soon becomes manifest that the style of reason ing which the student has heretofore employed must be abandoned. Examples of curves without tangents, of curves completely filling areas, and other strange configurations so familiar to the analyst of to-day, make it clear that the rough and ready reasoning which rests on geometric intuition must give way to a finer and more delicate analysis. It is necessary for him to learn to think in the e, 8 forms of Cauchy and Weierstrass.

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