Newtonian Mechanics is usually presented as a theory of point particles (and forces). My impression of the status of continuum mechanics is that it is mostly taken as an approximate description for certain situations, where many particles are present and where we are interested only in bulk motion. In situations like modeling a string, we see that the continuum limes leads to a very good approximation for the motion of pointlike masses connected by springs under certain conditions. This seems to point in the direction that continuum mechanics is an approximation to point particle mechanics.
But point particles should also be easy to accommodate as a special case in continuum mechanics by admitting delta distributions and the like. Taking this viewpoint, continuum mechanics seems to be a generalization of point particle mechanics.
Therefore my question: Is continuum mechanics a generalization or an approximation to point particle mechanics? Or can it be argued that both are equally valid starting points?
Part of my motivation to ask this question is that I sometimes have difficulties to connect concepts from one viewpoint to the other (see e.g. my question about the point of application of a force).