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Let $f$ be defined on $[a,b]$ and there uniformly differentiable ($\,$the $\delta$ in the definition of derivative is independent of the point).
Given $\epsilon>0$, choose a partition $P \, : \, a=a_0<a_1<\ldots<a_n=b \,$ of $ \,[a,b] \,$ with $||P|| \lt \delta$, and apply the definition to the points $\,a_0,\ldots,a_{n-1} \,$ getting $$\left|\frac {f(a_{i+1})-f(a_i)}{a_{i+1}-a_i}-f'(a_i) \right|<\epsilon\qquad(i=0,\ldots,n-1)$$ Typically, at an elementary level, two activities are possible by simple passages $\,$(removing the fraction and absolute value, summing over i both members, etc.).
One can prove the mean value inequality $\,$(only the Archimedean axiom is needed) $$\inf_{x \in [a,b]} f'(x) \le \frac {f(b)-f(a)}{b-a} \le \sup_{x \in [a,b]} f'(x)$$ One can prove also that $$\left |f(b)-f(a)-\sum_{i=0}^{n-1} f'(a_i)(a_{i+1}-a_i) \right |<\epsilon(b-a)$$ i.e. $\,f(b)-f(a) \,$ is the limit of a sequence of Cauchy left sums of $f'$.
The latter is a purely analytical motivation to Cauchy's proof of the existence of a primitive of a continuous function.
Why the concept is not generally developed in textbooks ?
Do you know other connected elementary statements ?

Jyrki Lahtonen
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Tony Piccolo
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  • Isn't "uniformly differentiable" equivalent to differentiable with continuous derivative? – 23rd Apr 24 '13 at 16:33
  • @Landscape yes ! – Tony Piccolo Apr 24 '13 at 16:39
  • Then I think this equivalence explains why this concept is not widely developed. – 23rd Apr 24 '13 at 16:40
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    Elementary real analysis texts take a different approach, using Dedekind's completeness axiom. Using that they prove, in particular, the Mean Value Theorem for all differentiable functions, not just uniformly differentiable ones. (And it's not overkill: MVT certainly implies Dedekind completeness.) I suppose that writers of the standard textbooks are quite familiar and happy with this approach so stick with it. Could you say more about what advantages you perceive in your approach? – Pete L. Clark May 04 '13 at 07:04
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    Also, I'd be appreciative of a (preferably modern, English language) reference to "Cauchy's proof of the existence of a primitive of a continuous function". I am not an expert on the history here, but I had the vague impression that Cauchy did not have a completely satisfactory proof of this. – Pete L. Clark May 04 '13 at 07:08
  • @Pete L. Clark $;$ see the following: it is an old story. – Tony Piccolo May 04 '13 at 14:39
  • @Pete L. Clark $;$ for Cauchy, you are right but the idea is his. – Tony Piccolo May 04 '13 at 14:47
  • $@$Tony: I followed your link but wasn't sure what I was supposed to see. – Pete L. Clark May 04 '13 at 16:07
  • @Pete $;$ references: by Cauchy's student and friend Abbé Moigno Lecons de calcul différentiel et de calcul intégral, volume 2 pp.1-9 $;$ T.M.Flett - Differential Analysis pp. 149-151. For a translation of Cauchy's papers, J.V.Grabiner The Origins of Cauchy's Rigorous Calculus pp. 171-175 – Tony Piccolo May 05 '13 at 07:50
  • @Pete $;$ e.g. you have to teach the above existence theorem (Cauchy's proof): start from a continuously differentiable function, show that it is uniformly differentiable by the mean value inequality, use a uniform partition of the interval and then find a convergent sequence of left sums of the derivative; now, motivated, start from a continuous function and try to show that a sequence of left sums is convergent by Cauchy criterion and so on, discovering that it works. There is no area to consider, the matter is purely analytical and the left sums are not pulled from out of the blue. – Tony Piccolo May 05 '13 at 09:58
  • @Tony: the main thing that I don't understand is your claim (if that is your claim) that this approach is somehow preferable to the standard one. What specific result is being given a better treatment this way: the existence of a primitive of an arbitrary continuous function? If I am understanding you correctly, you don't want to prove this using the Riemann integral. But why not? – Pete L. Clark May 05 '13 at 10:29
  • @Pete: sure, I want to avoid the formalism of tagged partitions and general Riemann sums. Pedagogically speaking, sometimes these things make the difference, if the target is limited. – Tony Piccolo May 05 '13 at 11:09
  • @Tony: Darboux's approach does avoid taggings of partitions and general Riemann sums. See for instance Chapter 8 of http://math.uga.edu/~pete/2400full.pdf, in which Riemann sums are not introduced until after the integrability of continuous functions. (Rudin's Principles never uses Riemann sums, except in the exercises.) Are you saying that the approach described in your question is simpler than Darboux's approach? Do you have it written out somewhere in full, so the two can be compared? – Pete L. Clark May 05 '13 at 11:24
  • @Pete: no, I should like to write a Riemann-free essay about antidifferentiation (or Newton integration so called by several authors), see my question. Coming back to Cauchy, I want to remain adherent to the spirit of Cauchy's proof. By the way I love Cauchy's criterion (those concentrated sequences ...). Darboux is out ! – Tony Piccolo May 05 '13 at 15:05

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