Would real analysis be complete without the fundamental theorem of calculus?
Michael Deveau, Holger Teismann · Elemente der Mathematik · 2015
Echoing L.R.Ford's opening words 1 of his delightful Monthly article [5], perhaps we, too, owe an apology to the reader for asking a seemingly flippant question in the title of this paper, whose answer must so obviously be 'no'.After all, the adjective "fundamental" says it all -even if, as Bressoud points out, that designation did not come into use until relatively recently [1].We admit that we chose the title for effect, accepting the possibility of leading the reader astray; a more descriptive title would have been: "would the real numbers be complete without the Fundamental Theorem of Calculus?"In some sense, however, the title is actually accurate in that this paper will show that a mathematical "world" (which we interpret to mean "totally ordered field") without the Fundamental Theorem of Calculus would necessarily be lacking of many of the most cherished parts of Real Analysis.Over the last decade or so it has been noticed that many statements / theorems from the standard canon of (single-variable) Real Analysis not only crucially depend on the completeness of the real numbers but are in fact equivalent to completeness; see [9,8,11] for various lists of such statements.While this fact may be reasonably well known, or at least be somewhat expected, for some statements such as the Intermediate, Mean, and Extreme Value Theorems, it may be more surprising for others, such as the Ratio Test [8], the Principle of Real Induction [3] or the Weierstrass Approximation Theorem.The most surprising feature of the list of statements equivalent to completeness, however, may very well be its sheer size, which, in its most recent version [4], comprises 70 items!Curiously, the most coveted candidate for the list, the Fundamental Theorem of Calculus (FTC), has been the most "difficult customer" in this enterprise and so far resisted inclusion 2 .