  We prove a single-value version of Reshetnyak's theorem. Namely, if a
non-constant map $f \in W^{1,n}_{\text{loc}}(\Omega, \mathbb{R}^n)$ from a
domain $\Omega \subset \mathbb{R}^n$ satisfies the estimate $\lvert Df(x)
\rvert^n \leq K J_f(x) + \Sigma(x) \lvert f(x) - y_0 \rvert^n $ for some $K
\geq 1$, $y_0\in \mathbb{R}^n$ and $\Sigma \in
L^{1+\varepsilon}_{\text{loc}}(\Omega)$, then $f^{-1}\{y_0\}$ is discrete, the
local index $i(x, f)$ is positive in $f^{-1}\{y_0\}$, and every neighborhood of
a point of $f^{-1}\{y_0\}$ is mapped to a neighborhood of $y_0$. Assuming this
estimate for a fixed $K$ at every $y_0 \in \mathbb{R}^n$ is equivalent to
assuming that the map $f$ is $K$-quasiregular, even if the choice of $\Sigma$
is different for each $y_0$. Since the estimate also yields a single-value
Liouville theorem, it hence appears to be a good pointwise definition of
$K$-quasiregularity. As a corollary of our single-value Reshetnyak's theorem,
we obtain a higher-dimensional version of the argument principle that played a
key part in the solution to the Calder\'on problem.
