I refer to a note by Lebesgue Remarques sur la définition de l'intégrale, Bull.Sci.Math. 29 (1905) 272-275 not very known (see pdf for an exposition in English).
It is a pedagogical note containing a method of proving the existence of an antiderivative of a continuous function alternative to Cauchy-Riemann integration (or a way, among others, to introduce the Newton integral Riemann-free).
In the first part Lebesgue shows that a continuous function admits an antiderivative because it can be represented by a uniformly convergent sequence of piecewise linear functions and one knows that a uniformly convergent sequence of derivatives represents a derivative.
I doubt that such a thing was shown for the first time in 1905 ...
Do you know something about ?