# These unknown uniformly differentiable functions

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 ?

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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
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 4 '13 at 7:04
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 4 '13 at 7:08