When differential and integral calculus were first discovered, in the 1600-1700s, they were proven to be immensely useful in so many applications it is almost mind boggling. But as far as I know, it was first long into the 1900s before any strict theoretic foundation for infinitesimals was actually established.
In many modern physics and engineering applications, concepts such as multiresolution analysis, scale spaces and coarse-to-fine play an important role. Roughly speaking in those concepts short time (or length) integrals and differential operators are mixed together. Which size portion of each decides the scale / resolution.
Is there some connection between how infinitesimals were first formally defined and these developments?