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On the Teaching of Analysis

  • E. G. Phillips

Every attempt to give an elementary rigorous course of Analysis is beset by several unsurmountable difficulties. Every teacher of the subject will readily agree that the two main obstacles are (i) Dedekind’s theorem, and (ii) the Heine-Borel theorem. Of these the first may be rendered less apparent, for in an elementary treatment there is no great objection to the postulation of the existence of an upper and a lower bound for every bounded linear set of numbers. The second cannot, however, be avoided, for the important theorems of the differential calculus, such as Rolle’s theorem, the Mean-value theorem, Cauchy’s formula, and Taylor’s theorem can only be rigorously proved by appealing to certain properties of continuous functions, which in their turn depend upon the Heine-Borel theorem, or upon some alternative simpler theorem on intervals such as the one which Carslaw uses to replace it.

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page 571 note * Theory of Fourier Series and Integrals (1921), § 18.

page 571 note See, for example, Gibson’s Calculus (1919), pp. 77 and 129.

page 571 note The order of development of the subject which I prefer will be seen in a book which I have just completed, and which will shortly be published by the Cambridge University Press.

page 572 note * In 1931 this will be superseded In Bangor by a joint Honours course in Pure and Applied mathematics, and so a good deal of the work now done in Pure mathematics may have to be abandoned.

page 572 note See Introduction to Mathematical Philosophy, p. 72.

page 573 note * Pure Mathematics (1925), p. 389.

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The Mathematical Gazette
  • ISSN: 0025-5572
  • EISSN: 2056-6328
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