The Image of a Weakly Differentiable Mapping

Let $\Omega \subset {\mathbf R}^n$ be an open ball, $n \ge 2$. Suppose that $f,g : \Omega \to {\mathbf R}^n$, $f = g$ on $\partial\Omega$, and that f is injective. In case f and g are continuous, then $f(\overline{\Omega}) \subset g(\overline{\Omega})$. We extend this result to generally discontinuous mappings belonging to suitable Sobolev spaces under appropriate notions of injectivity and boundary equality.