I How do the concepts of "boundary"and"infinitesimal" interact

It is all a question of scale. For example, if you are concerned with how a ball of radius r bounces off a sidewalk that is more or less flat, with perturbations up to +/- 1/r^10, those small perturbations will not likely impact the trajectory of the bouncing ball very much. If you are looking to get to that level of precision, you would probably also need to account for other physical influences as well. If you want to account for all the additional factors, you can work through the math to see what the maximum impact would be. In most cases, it is negligible.

Based on the scale of your problem and desired precision, you can assume away most of those questions of granularity. Clearly, if you are working with lasers in the optical frequencies, your definition of smooth will be very different from that of someone looking into bouncing a soccer ball on concrete.

 

Why must one have a boundary that is perfectly sharp in order to have a boundary at all? Where, exactly, is the boundary between the trunk of a tree and its roots?

 

What is the question?