 Bounded isolated-point removability for weak first derivatives

A bounded function which is $C^1$ away from one point in dimension $n\ge 2$, and whose classical gradient is locally integrable across that point, has that gradient as its weak derivative. Cut a test function off within distance $\epsilon$ of the point. The error from differentiating the cutoff is bounded by $C\Vert u{\Vert }_{\infty }{\epsilon }^{n-1}$ and vanishes. The other terms converge by local integrability. In dimension one the Heaviside step function is a counterexample: its classical derivative off the point is zero, but its distributional derivative is a Dirac delta.