Flux around a removable point singularity ($\left|{\int }_{\partial B_{\epsilon }}\mathbf{F}\cdot \mathbf{n}dS\right|=O({\epsilon }^{d-1+\alpha })$)

A continuous extension of a vector field across one point need not be differentiable, so it does not automatically meet the classical hypotheses of the divergence theorem. Instead apply the divergence theorem after deleting a ball $B_{\epsilon }$ around that point. If $\Vert \mathbf{F}\Vert \le C{\epsilon }^{\alpha }$ on its boundary in $d$ dimensions, the magnitude of the inner flux is at most a constant times ${\epsilon }^{d-1+\alpha }$. When $d-1+\alpha >0$ this tends to zero. If the divergence is also integrable, passing to the limit recovers the outer flux from the volume integral. For example, a planar vector field with $\Vert \mathbf{F}\Vert \le r$ has an inner flux of order ${\epsilon }^2$, even when its extension is not differentiable at the origin.