  This paper is concerned with developing a theory of traces for functions that
are integrable but need not possess any differentiability within their domain.
Moreover, the domain can have an irregular boundary with cusp-like features and
codimension not necessarily equal to one, or even an integer. Given
$\Omega\subseteq\mathbb{R}^n$ and $\Gamma\subseteq\partial\Omega$, we introduce
a function space $\mathscr{N}^{s(\cdot),p}(\Omega)\subseteq
L^p_{\text{loc}}(\Omega)$ for which a well-defined trace operator can be
identified. Membership in $\mathscr{N}^{s(\cdot),p}(\Omega)$ constrains the
oscillations in the function values as $\Gamma$ is approached, but does not
imply any regularity away from $\Gamma$. Under connectivity assumptions between
$\Omega$ and $\Gamma$, we produce a linear trace operator from
$\mathscr{N}^{s(\cdot),p}(\Omega)$ to the space of measurable functions on
$\Gamma$. The connectivity assumptions are satisfied, for example, by all
$1$-sided nontangentially accessible domains. If $\Gamma$ is upper
Ahlfors-regular, then the trace is a continuous operator into a
Sobolev-Slobodeckij space. If $\Gamma=\partial\Omega$ and is further assumed to
be lower Ahlfors-regular, then the trace exhibits the standard Lebesgue point
property. To demonstrate the generality of the results, we construct
$\Omega\subseteq\mathbb{R}^2$ with a $t>1$-dimensional Ahlfors-regular
$\Gamma\subseteq\partial\Omega$ satisfying the main domain hypotheses, yet
$\Gamma$ is nowhere rectifiable and for every neighborhood of every point in
$\Gamma$, there exists a boundary point within that neighborhood that is only
tangentially accessible.
