Removable sets for the flux of continuous vector fields

We show that any closed set $E$ having a $\sigma$-finite $(n-1)$-dimensional Hausdorff measure does not support the nonzero distributional divergence of a continuous vector field; in particular it has the property that any $C^1$ function in $\mathbb{R}^n$ that is harmonic outside it is harmonic in $\mathbb{R}^n$. We also exhibit a compact set $E$ having Hausdorff dimension $n-1$, supporting the nonzero distributional divergence of a continuous vector field yet having the property that any $C^1$ function that is harmonic outside $E$ is harmonic in $\mathbb{R}^n$.