Consider a function f continuous on a compact interval.

Approximate it by a sequence of polygonal functions (you can).

Then consider a sequence of primitives of the polygonal functions (you can).

At last consider the limit of the latter sequence (you can).

Now you have found a primitive of f (you know) without integration.

This is the content of the first part of a not very known note by Lebesgue Remarques sur la définition de l'intégrale, Bull.Sci.Math. 29 (1905) 272-275 (see pdf for an exposition in English).

I doubt that such a thing was shown for the first time in 1905.

Lebesgue's good faith is beyond discussion of course.

Do you know something about ?

  • $\begingroup$ The trapezoid rule? $\endgroup$ – Noah Stein Apr 16 '13 at 20:58
  • $\begingroup$ @Noah Stein yes! $\endgroup$ – Antonio Piciulin Apr 16 '13 at 21:05
  • 1
    $\begingroup$ This is basically how I think about the fundamental theorem... $\endgroup$ – Steve Apr 16 '13 at 21:11
  • $\begingroup$ To avoid misunderstanding: quite a long time several authors call antidifferentiation Newton integration. This allowed me the pun of the title. The second integration means Cauchy-Riemann integration of course. $\endgroup$ – Antonio Piciulin Apr 17 '13 at 7:45
  • $\begingroup$ Anyhow the procedure shows the existence of a primitive directly avoiding the typical tour (i) develop the Riemann integral (ii) prove that continuous functions are Riemann integrable (iii) prove the fundamental theorem of the calculus for the Riemann integral. Obviously one can use step functions to approximate f. $\endgroup$ – Antonio Piciulin Apr 19 '13 at 7:28
  1. What does this procedure have to do with Newton? Just curious.

  2. If you approximate with "polygonal" (=piecewise linear) functions in the most natural way, that is take points $(x_k,f(x_k))$ and connect them with straight line segments, what you obtain is the "trapezoid rule" for approximate evaluation of integrals.

  3. Of course, the trapezoid rule is "a theorem of Adam"; already Gauss knew much more sophisticated rules.

  4. Convergence of the procedure with any continuous function and any (reasonable) choice of $x_k$ is a trivial exercise for modern students, and it was probably in 1905. I looked at the paper, and it seems to me that in it, Lebesgue only proposes a simple way to TEACH the integral. There is nothing really new in this paper.

  • $\begingroup$ for Newton see R.Henstock - Lectures on the theory of integration page 29 but the possible quotations are many $\endgroup$ – Antonio Piciulin Apr 16 '13 at 21:19
  • $\begingroup$ for me the fact that a uniformly convergent sequence of derivatives represents a derivative is not Adam's $\endgroup$ – Antonio Piciulin Apr 17 '13 at 8:31
  • $\begingroup$ Antonio, In this procedure you only need that integrals of uniformly convergent functions are convergent. $\endgroup$ – Alexandre Eremenko Apr 17 '13 at 11:54
  • $\begingroup$ In the end I agree with you: it's the meaning of my question. But Medvedev ... ? $\endgroup$ – Antonio Piciulin Apr 17 '13 at 15:33
  • $\begingroup$ What about Medvedev? Does he say anything about this paper of Lebesgue? $\endgroup$ – Alexandre Eremenko Apr 18 '13 at 3:53

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