  We study continuity properties of Sobolev mappings $f \in
W_{\mathrm{loc}}^{1,n} (\Omega, \mathbb{R}^n)$, $n \ge 2$, that satisfy the
following generalized finite distortion inequality \[\lvert Df(x)\rvert^n \leq
K(x) J_f(x) + \Sigma (x)\] for almost every $x \in \mathbb{R}^n$. Here $K
\colon \Omega \to [1, \infty)$ and $\Sigma \colon \Omega \to [0, \infty)$ are
measurable functions. Note that when $\Sigma \equiv 0$, we recover the class of
mappings of finite distortion, which are always continuous. The continuity of
arbitrary solutions, however, turns out to be an intricate question. We fully
solve the continuity problem in the case of bounded distortion $K \in L^\infty
(\Omega)$, where a sharp condition for continuity is that $\Sigma$ is in the
Zygmund space $\Sigma \log^\mu(e + \Sigma) \in L^1_{\mathrm{loc}}(\Omega)$ for
some $\mu > n-1$. We also show that one can slightly relax the boundedness
assumption on $K$ to an exponential class $\exp(\lambda K) \in
L^1_{\mathrm{loc}}(\Omega)$ with $\lambda > n+1$, and still obtain continuous
solutions when $\Sigma \log^\mu(e + \Sigma) \in L^1_{\mathrm{loc}}(\Omega)$
with $\mu > \lambda$. On the other hand, for all $p, q \in [1, \infty]$ with
$p^{-1} + q^{-1} = 1$, we construct a discontinuous solution with $K \in
L^p_{\mathrm{loc}}(\Omega)$ and $\Sigma/K \in L^q_{\mathrm{loc}}(\Omega)$,
including an example with $\Sigma \in L^\infty_{\mathrm{loc}}(\Omega)$ and $K
\in L^1_{\mathrm{loc}}(\Omega)$.
