These unknown uniformly differentiable functions

Question

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 ?