  In this paper we connect Calder\'on and Zygmund's notion of $L^p$\-
-differentiability with some recent characterizations of Sobolev spaces via the
asymptotics of non-local functionals due to Bourgain, Brezis, and Mironescu. We
show how the results of the former can be generalized to the setting of the
latter, while the latter results can be strengthened in the spirit of the
former. As a consequence of these results we give several new characterizations
of Sobolev spaces, a novel condition for whether a function of bounded
variation is in the Sobolev space $W^{1,1}$, and complete the proof of a
characterization of the Sobolev spaces claimed in the paper "Characterization
of Sobolev and BV spaces".
