  In this note, we introduce a variant of Calder\'on and Zygmund's notion of
$L^p$-differentiability - an \emph{$L^p$-Taylor approximation}. Our first
result is that functions in the Sobolev space $W^{1,p}(\mathbb{R}^N)$ possess a
first order $L^p$-Taylor approximation. This is in analogy with Calder\'on and
Zygmund's result concerning the $L^p$-differentiability of Sobolev functions.
In fact, the main result we announce here is that the first order $L^p$-Taylor
approximation characterizes the Sobolev space $W^{1,p}(\mathbb{R}^N)$, and
therefore implies $L^p$-differentiability. Our approach establishes connections
between some characterizations of Sobolev spaces due to Swanson using
Calder\'on-Zygmund classes with others due to Bourgain, Brezis, and Mironescu
using nonlocal functionals with still others of the author and Mengesha using
nonlocal gradients. That any two characterizations of Sobolev spaces are
related is not surprising, however, one consequence of our analysis is a simple
condition for determining whether a function of bounded variation is in a
Sobolev space.
