This is a reference request prompted by some intriguing comments made by Felix Klein.
In 1908, Felix Klein formulated a criterion of what it would take for a theory of infinitesimals to be successful. Namely, one must be able to prove a mean value theorem (MVT) for arbitrary intervals, including infinitesimal ones:
The question naturally arises whether ... it would be possible to modify the traditional foundations of infinitesimal calculus, so as to include actually infinitely small quantities in a way that would satisfy modern demands as to rigor; in other words, to construct a non-Archimedean system. The first and chief problem of this analysis would be to prove the mean-value theorem $$ f(x+h)-f(x)=h \cdot f'(x+\vartheta h) $$ from the assumed axioms. I will not say that progress in this direction is impossible, but it is true that none of the investigators have achieved anything positive.
This comment appears on page 219 in the book (Klein, Felix Elementary mathematics from an advanced standpoint. Arithmetic, algebra, analysis) originally published in 1908 in German.
Question 1. When Klein writes that none of the current investigators have achieved, etc., who is he referring to? There were a number of people working "in this direction" at the time, and it would be interesting to know whose work Klein had in mind: Stolz, Paul du Bois-Raymond (somewhat earlier), Hahn, Hilbert, etc.
Question 2. Did Klein elaborate in this direction in other works of his?
Question 3. Why is Klein focusing specifically on the mean value theorem? Was this theorem a preoccupation of other mathematicians at the time?