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David M. Bressoud Macalester College, St. Paul, MN Project NExT-WI, October 5, 2006 ―The task of the educator is to make the child‘s spirit pass again where its forefathers have gone, moving rapidly through certain stages but suppressing none of them. In this regard, the history of science must be our guide.‖ Henri Poincaré 1. Cauchy and uniform convergence 2. The Fundamental Theorem of Calculus 3. The Heine–Borel Theorem 1. Cauchy and uniform convergence 2. The Fundamental Theorem of Calculus 3. The Heine–Borel Theorem A Radical Approach to Real Analysis, 2nd edition due January, 2007 A Radical Approach to Lebesgue’s Theory of Integration, due December, 2007 ―What Weierstrass — Cantor — did was very good. That's the way it had to be done. But whether this corresponds to what is in the depths of our consciousness is a very Nikolai Luzin different question … … I cannot but see a stark contradiction between the intuitively clear fundamental formulas of the integral calculus and the incomparably artificial and Nikolai Luzin complex work of the ‗justification‘ and their ‗proofs‘. Cauchy, Cours d’analyse, 1821 ―…explanations drawn from algebraic technique … cannot be considered, in my opinion, except as heuristics that will sometimes suggest the truth, but which accord little with the accuracy that is so praised in the mathematical sciences.‖ Niels Abel (1826): ―Cauchy is crazy, and there is no way of getting along with him, even though right now he is the only one who knows how mathematics should be done. What he is doing is excellent, but very confusing.‖ Cauchy, Cours d’analyse, 1821, p. 120 Theorem 1. When the terms of a series are functions of a single variable x and are continuous with respect to this variable in the neighborhood of a particular value where the series converges, the sum S(x) of the series is also, in the neighborhood of this particular value, a continuous function of x. Abel, 1826: ―It appears to me that this theorem suffers exceptions.‖ x depends on n n depends on x “If even Cauchy can make a mistake like this, how am I supposed to know what is correct?” What is the Fundamental Theorem of Calculus? Why is it fundamental? The Fundamental Theorem of Calculus (evaluation part): If then Differentiate then Integrate = original fcn (up to constant) The Fundamental Theorem of Calculus (antiderivative part): If f is continuous, then Integrate then Differentiate = original fcn The Fundamental Theorem of Calculus (evaluation part): If then Differentiate then Integrate = original fcn (up to constant) The Fundamental Theorem of Calculus (antiderivative part): If f is continuous, then Integrate then Differentiate = original fcn 1820, The fundamental proposition of the theory of definite integrals: Siméon Poisson If then 1820, The fundamental proposition of the theory of definite integrals: Siméon Poisson If then Definite integral, defined as difference of antiderivatives at endpoints, is sum of products, f(x) times infinitesimal dx. Cauchy, 1823, first explicit definition of definite integral as limit of sum of products mentions the fact that en route to his definition of the indefinite integral. Earliest reference to Fundamental Theorem of the Integral Calculus is by Paul du Bois-Reymond (1880s). Popularized in English by E. W. Hobson: The Theory of Functions of a Real Variable, 1907 Granville (w/ Smith) Differential and Integral Calculus (starting with 1911 ed.), FTC: definite integral can be used to evaluate a limit of a sum of products. William A. Granville The real FTC: There are two distinct ways of viewing integration: • As a limit of a sum of products (Riemann sum), •As the inverse process of differentiation. The power of calculus comes precisely from their equivalence. Riemann‘s habilitation of 1854: Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe Purpose of Riemann integral: 1. To investigate how discontinuous a function can be and still be integrable. Can be discontinuous on a dense set of points. 2. To investigate when an unbounded function can still be integrable. Introduce improper integral. Riemann’s function: At the function jumps by The Fundamental Theorem of Calculus (antiderivative part): If f is continuous, then Integrate then Differentiate = original fcn This part of the FTC does not hold at points where f is not continuous. The Fundamental Theorem of Calculus (evaluation part): If then Differentiate then Integrate = original fcn (up to constant) Volterra, 1881, constructed function with bounded derivative that is not Riemann integrable. Vito Volterra Perfect set: equals its set of limit points Nowhere dense: every interval contains subinterval with no points of the set Non-empty, nowhere dense, perfect set described by H.J.S. Smith, 1875 Perfect set: equals its set of limit points Nowhere dense: every interval contains subinterval with no points of the set Non-empty, nowhere dense, perfect set described by H.J.S. Smith, 1875 Then by Vito Volterra, 1881 Perfect set: equals its set of limit points Nowhere dense: every interval contains subinterval with no points of the set Non-empty, nowhere dense, perfect set described by H.J.S. Smith, 1875 Then by Vito Volterra, 1881 Finally by Georg Cantor, 1883 Perfect set: equals its set of limit points Nowhere dense: every interval contains subinterval with no points of the set Non-empty, nowhere dense, perfect set described by H.J.S. Smith, 1875 Then by Vito Volterra, 1881 Finally by Georg Cantor, 1883 Volterra’s construction: Start with the function Restrict to the interval [0,1/8], except find the largest value of x on this interval at which F '(x) = 0, and keep F constant from this value all the way to x = 1/8. Volterra’s construction: To the right of x = 1/8, take the mirror image of this function: for 1/8 < x < 1/4, and outside of [0,1/4], define this function to be 0. Call this function . Now we slide this function over so that the portion that is not identically 0 is in the interval [3/8,5/8], that middle piece of length 1/4 taken out of the SVC set. We do the same thing for the interval [0,1/16]. We slide one copy of into each interval of length 1/16 that was removed from the SVC set. Volterra’s function, V satisfies: 1. V is differentiable at every x, V' is bounded. 2. For a in SVC set, V'(a) = 0, but there are points arbitrarily close to a where the derivative is +1, –1. V' is not Riemann integrable on [0,1] The Fundamental Theorem of Calculus (evaluation part): If then Differentiate then Integrate = original fcn (up to constant) Volterra, 1881, constructed function with bounded derivative that is not Riemann integrable. FTC does hold if we restrict f to be continuous or if we use Vito Volterra the Lebesgue integral and F is absolutely continuous. Lessons: 1. Riemann‘s definition is not intuitively natural. Students think of integration as inverse of differentiation. Cauchy definition is easier to comprehend. Lessons: 1. Riemann‘s definition is not intuitively natural. Students think of integration as inverse of differentiation. Cauchy definition is easier to comprehend. 2. Emphasize FTC as connecting two very different ways of interpreting integration. Go back to calling it the Fundamental Theorem of Integral Calculus. Lessons: 1. Riemann‘s definition is not intuitively natural. Students think of integration as inverse of differentiation. Cauchy definition is easier to comprehend. 2. Emphasize FTC as connecting two very different ways of interpreting integration. Go back to calling it the Fundamental Theorem of Integral Calculus. 3. Need to let students know that these interpretations of integration really are different. Heine–Borel Theorem Any open cover of a closed and bounded set of real numbers has a Eduard Heine finite subcover. Émile Borel 1821–1881 1871–1956 Heine–Borel Theorem Any open cover of a closed and bounded set of real numbers has a Eduard Heine finite subcover. Émile Borel 1821–1881 1871–1956 Due to Lebesgue, 1904; stated and proven by Borel for countable covers, 1895; Heine had very little to do with it. P. Dugac, ―Sur la correspondance de Borel …‖ Arch. Int. Hist. Sci.,1989. 1852, Dirichlet proves that a continuous function on a closed, bounded interval is uniformly continuous. The proof is very similar to Borel and Lebesgue‘s proof of Heine–Borel. 1872, Heine reproduces this proof without attribution to Dirichlet in ―Die Elemente der Functionenlehre‖ 1872, Heine reproduces this proof without attribution to Dirichlet in ―Die Elemente der Functionenlehre‖ Weierstrass,1880, if a series converges uniformly in some open neighborhood of every point in [a,b], then it converges uniformly over [a,b]. 1872, Heine reproduces this proof without attribution to Dirichlet in ―Die Elemente der Functionenlehre‖ Weierstrass,1880, if a series converges uniformly in some open neighborhood of every point in [a,b], then it converges uniformly over [a,b]. Pincherle,1882, if a function is bounded in some open neighborhood of every point in [a,b], then it is bounded over [a,b]. Harnack, 1885, considered the question of the ―measure‖ of an arbitrary set. Considered and rejected the possibility of using countable collection of open intervals. Axel Harnack 1851–1888 Harnack, 1885, considered the question of the ―measure‖ of an arbitrary set. Considered and rejected the possibility of using countable collection of open intervals. Axel Harnack 1851–1888 Harnack, 1885, considered the question of the ―measure‖ of an arbitrary set. Considered and rejected the possibility of using countable collection of open intervals. Axel Harnack 1851–1888 Harnack assumed that the complement of a countable union of intervals is a countable union of intervals, in which case the answer is YES. Harnack, 1885, considered the question of the ―measure‖ of an arbitrary set. Considered and rejected the possibility of using countable collection of open intervals. Axel Harnack 1851–1888 Harnack assumed that the complement of a countable union of intervals is a countable union of intervals, in which case the answer is YES. Cantor’s set: 1883 Borel, 1895 (doctoral thesis, 1894), problem of analytic continuation across a boundary on which lie a countable dense set of poles Arthur Schönflies, 1900, claimed Borel‘s result also holds for uncountable covers, pointed out connection to Heine‘s proof of uniform continuity. First to call this the Heine–Borel theorem. Arthur Schönflies, 1900, claimed Borel‘s result also holds for uncountable covers, pointed out connection to Heine‘s proof of uniform continuity. First to call this the Heine–Borel theorem. 1904, Henri Lebesgue to Borel, ―Heine says nothing, nothing at all, not even remotely, about your theorem.‖ Suggests calling it the Borel– Schönflies theorem. Proves the Schönflies claim that it is valid for uncountable covers. Arthur Schönflies, 1900, claimed Borel‘s result also holds for uncountable covers, pointed out connection to Heine‘s proof of uniform continuity. First to call this the Heine–Borel theorem. 1904, Henri Lebesgue to Borel, ―Heine says nothing, nothing at all, not even remotely, about your theorem.‖ Suggests calling it the Borel– Schönflies theorem. Proves the Schönflies claim that it is valid for uncountable covers. Paul Montel and Giuseppe Vitali try to change designation to Borel–Lebesgue. Borel in Leçons sur la Théorie des Fonctions calls it the ―first fundamental theorem of measure theory.‖ Lessons: 1. Heine–Borel is far less intuitive than other equivalent definitions of completeness. Lessons: 1. Heine–Borel is far less intuitive than other equivalent definitions of completeness. 2. In fact, Heine–Borel can be counter-intuitive. Lessons: 1. Heine–Borel is far less intuitive than other equivalent definitions of completeness. 2. In fact, Heine–Borel can be counter-intuitive. 3. Heine–Borel lies at the root of Borel (and thus, Lebesgue) measure. This is the moment at which it is needed. Much prefer Borel‘s name: First Fundamental Theorem of Measure Theory. This PowerPoint presentation is available at www.macalester.edu/~bressoud/talks A draft of A Radical Approach to Lebesgue’s Theory of Integration is available at www.macalester.edu/~bressoud/books