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# A story from the Development of Real Analysis

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These slides will be available at www.macalester.edu/~bressoud/talks Moravian College February 20, 2009 David Bressoud Macalester College St. Paul, MN“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 1854–1912Series, continuity, differentiation 1800–1850 Integration, structure of the real numbers 1850–1910“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 different question … Nikolai Luzin 1883–1950… I cannot but see a stark contradiction between the intuitively clear fundamental formulas of the integral calculus and the incomparably artificial and complex work of the ‘justification’ and their ‘proofs’. Nikolai Luzin 1883–1950What is the Fundamental Theorem of Calculus? Why is it fundamental?If then !f x dx F b F aab!"#!"!!"".F x f x',()=()The Fundamental Theorem of Calculus (evaluation part):!Differentiate then Integrate  original fcn (up to constant) Integrate then Differentiate  original fcn The Fundamental Theorem of Calculus (antiderivative part):!ddtf x( )0t∫dx = f t( ).If f is continuous, then !The Fundamental Theorem of Calculus (antiderivative part):!ddtf x( )0t∫dx = f t( ).If f is continuous, then !If then !f x dx F b F aab!"#!"!!"".F x f x',()=()The Fundamental Theorem of Calculus (evaluation part):!Differentiate then Integrate = original fcn (up to constant) Integrate then Differentiate  original fcnCauchy, 1823, first explicit definition of definite integral as limit of sum of products mentions the fact that ddtf x( )0t∫dx = f t( )en route to his definition of the indefinite integral. f x( )dx = limn→∞ab∫f xi −1( )i =1n∑xi− xi −1( );1828–1923 Fundamental Theorem for Integrals De L’Analyse Infinitésimal, Charles de Freycinet, 1860Fundamental Theorem for Integrals De L’Analyse Infinitésimal, Charles de Freycinet, 1860 Fundamental Theorem of Integral Calculus used by Paul du Bois-Reymond in appendix to paper on trigonometric series, 1876. 1828–1923 1831–1889Fundamental Theorem of the Integral Calculus popularized in English by E. W. Hobson, The Theory of Functions of a Real Variable, 1907. 1856–1933Fundamental Theorem of the Integral Calculus popularized in English by E. W. Hobson, The Theory of Functions of a Real Variable, 1907. 1856–1933 1877–1947 G. H. Hardy, 2nd edition of A Course of Pure Mathematics, 1914, refers to it as the Fundamental Theorem of Calculus.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. limmax Δxi→0f xi*( )i =1n∑Δxi1826–1866Darboux’s equivalent definition: Given function f bounded on [a,b] and partition P: a = x0 < x1 < x2 < … < xn = b, Upper Darboux sum: Lower Darboux sum: Gaston Darboux 1842–1917 sup f xi*( )( )i =1n∑Δxiinf f xi*( )( )i =1n∑Δxif is Riemann integrable on [a,b] if and only if for each ε > 0 there is a partition P so that the difference between the upper and lower Darboux sums is less than ε: sup f xi*( )− inf f xi*( )( )i =1n∑Δxi<ε.Riemann’s function:!xx!"#!\$%&"#\$%\$nearest integer , when this is when distance to nearest integer is 12012,,fxnxnn!"#\$%#!"21At the function jumps by !xabab=()=221, gcd , ,!228bRiemann’s function:!fxnxnn!"#\$%#!"21F t( )=nx{ }n2n =1∞∑0t∫=1n2n =1∞∑nx{ }dx0t∫F is an anti-derivative. Does ddtF t( )= f t( )?No! Darboux showed that any function that is a derivative must satisfy the intermediate value property (the function cannot have any vertical jumps). Riemann’s function f has a vertical jump at every rational number with an even denominator.Integrate then Differentiate  original function The Fundamental Theorem of Calculus (antiderivative part):!ddtf x( )0t∫dx = f t( ).If f is continuous, then !This part of the FTC does not hold at points where f is not continuous.If then !f x dx F b F aab!"#!"!!"".F x f x',()=()The Fundamental Theorem of Calculus (evaluation part):!Differentiate then Integrate  original fcn (up to constant) Does this work for all differentiable functions F ? No! F x( )= x ,dFdx=12 xfor x > 0.12 x is not bounded, and so is not Riemann integrable on [0,1].If then !f x dx F b F aab!"#!"!!"".F x f x',()=()The Fundamental Theorem of Calculus (evaluation part):!Differentiate then Integrate  original fcn (up to constant) What if the function F has a finite derivative at every point in [a,b]? No! F x( )= x2sin x−2( ),dFdx=2x sin x−2( )− 2x−1cos x−2( ), for x ≠ 0,0, for x = 0.⎧⎨⎪⎩⎪2x sin x−2( )− 2x−1cos x−2( ) is not bounded, and so is not Riemann integrable.What if the function F has a finite derivative at every point in [a,b] and the derivative stays bounded? If then !f x dx F b F aab!"#!"!!"".F x f x',()=()The Fundamental Theorem of Calculus (evaluation part):!Differentiate then Integrate  original fcn (up to constant)If then !f x dx F b F aab!"#!"!!"".F x f x',()=()The Fundamental Theorem of Calculus (evaluation part):!Differentiate then Integrate  original fcn (up to constant) Volterra, 1881, constructed function with bounded derivative that is not Riemann integrable. Vito Volterra 1860–1940Fxxxxx!"#!"!#"#\$21000sin , ,,.Fxxxxx!"#!"!#"#\$21000sin , ,,.F x xxxx' sin cos , .!"#!"!!""2110Fxxxxx!"#!"!#"#\$21000sin , ,,.F x xxxx' sin cos , .!"#!"!!""2110FF h Fhhhhhhhhh' limlimsinlim sin .001100020!"#!"!!"#!"#!"#"""Fxxxxx!"#!"!#"#\$21000sin , ,,.F x xxxx' sin cos , .!"#!"!!""2110FF h Fhhhhhhhhh' limlimsinlim sin