The main goal of this article is to show that there are qualitative and quantitative differences between continuum fields obtained by the coarse‐graining of systems comprising discrete constituents at finite or very large (“infinite”) resolution. This is demonstrated by two examples: the stress field is symmetric for infinite coarse‐graining scales but can be asymmetric for finite resolution; the classical expression for the elastic energy density needs to be corrected by a term that vanishes in the limit of infinite coarse‐graining scale. The article also presents a brief and somewhat biased introduction to the subject of coarse‐graining (or homogenization or averaging) and a summary of part of the recent and not‐so‐recent work by the author and some of his collaborators. It is shown that it is possible to coarse grain discrete systems in such a way that the fields are smooth and the resolution controlled. In particular, the use of smooth coarse graining functions of pre‐determined resolution ensures tha...