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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 ?

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  • $\begingroup$ The trapezoid rule? $\endgroup$
    – Noah Stein
    Apr 16, 2013 at 20:58
  • $\begingroup$ @Noah Stein yes! $\endgroup$ Apr 16, 2013 at 21:05
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    $\begingroup$ This is basically how I think about the fundamental theorem... $\endgroup$
    – Steve
    Apr 16, 2013 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. $\endgroup$ Apr 17, 2013 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. $\endgroup$ Apr 19, 2013 at 7:28

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  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.

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  • $\begingroup$ for Newton see R.Henstock - Lectures on the theory of integration page 29 but the possible quotations are many $\endgroup$ Apr 16, 2013 at 21:19
  • $\begingroup$ for me the fact that a uniformly convergent sequence of derivatives represents a derivative is not Adam's $\endgroup$ Apr 17, 2013 at 8:31
  • $\begingroup$ Antonio, In this procedure you only need that integrals of uniformly convergent functions are convergent. $\endgroup$ Apr 17, 2013 at 11:54
  • $\begingroup$ In the end I agree with you: it's the meaning of my question. But Medvedev ... ? $\endgroup$ Apr 17, 2013 at 15:33
  • $\begingroup$ What about Medvedev? Does he say anything about this paper of Lebesgue? $\endgroup$ Apr 18, 2013 at 3:53

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