= Continuous weak gradient gives a C1 representative
{title2=$D_ju\in C\ \Longrightarrow\ u=U\text{ a.e. with }U\in C^1$}
If a locally integrable function has <weak derivatives> represented by a continuous <vector field> $g$, it has a representative $U\in C^1$ with $\nabla U=g$. The assertion is local, so work on a ball compactly contained in the domain and regularize using a <mollifier>. The regularizations satisfy the segment identity $u_\varepsilon(x)-u_\varepsilon(y)=\int_0^1\nabla u_\varepsilon(y+t(x-y))\cdot(x-y)\,dt$. Averaging over a smaller ball in $y$ fixes the additive constant through local $L^1$ convergence. The gradients converge uniformly on compact sets, so the averaged identity gives local uniform convergence to $U$. Passing to the limit on each segment then gives $U(x+h)-U(x)-g(x)\cdot h=o(|h|)$. Continuity of $g$ proves that $U$ is continuously <differentiable>.
Back to article page
