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Recall that $g_i\in L_{\mathrm{loc}}^1(\Omega )$ is the weak derivative $D_{i}u$ if $u\in L_{\mathrm{loc}}^1(\Omega )$ and $\int u{D}_i\varphi =-\int g_i\varphi$ for all $\varphi \in C_c^{\infty }(\Omega )$. A global weak derivative restricts to every open neighborhood simply by taking test functions supported there.

Conversely, suppose local weak derivatives exist near each point. local integrability of $u$ follows by a finite cover of each compact subset. On overlapping neighborhoods the two candidates for $D_{i}u$ have identical integrals against every smooth compactly supported test function. The fundamental lemma for locally integrable functions therefore makes them equal almost everywhere. Choose a countable local cover, possible in Euclidean space; the countably many exceptional overlap sets can all be discarded. The derivatives consequently patch to measurable $g_i\in L_{\mathrm{loc}}^1(\Omega )$.

For a test function $\varphi$, take a smooth partition of unity $({\chi }_j)$ subordinate to the local cover and locally finite on its support. Only finitely many terms occur there. Because ${\sum }_j{\chi }_j=1$ near the support,The derivatives of the partition sum to zero, explaining the first equality. Thus the patched functions are the global weak derivatives. This proves both directions of locality of weak differentiability.

By locality of weak differentiability, it is enough to work near the exceptional point; translate it to zero. Let $g_i$ denote the classical derivative off zero, assigning any value at zero. For $\varphi \in C_c^{\infty }(\Omega )$ choose a smooth cutoff ${\chi }_{\epsilon }$ which is zero on $B_{\epsilon }$, one outside $B_{2\epsilon }$, satisfies $|D{\chi }_{\epsilon }|\le C/\epsilon$, and has its transition region contained in $\Omega$. The product $\varphi {\chi }_{\epsilon }$ avoids the singular point, so classical integration by parts givesThe middle term has absolute value at mostwhen $n\ge 2$. The other terms tend to $\int u{D}_i\varphi$ and $-\int g_i\varphi$, since $u$ is locally integrable and $g_i\in L_{\mathrm{loc}}^1$. Therefore $\boxed{D_{i}u=g_i\text{ weakly on all of }\Omega }$. Its value at the single point is irrelevant. This is bounded isolated-point removability for weak first derivatives.

For $n=1$, take $u(x)=0$ for $x<0$ and $u(x)=1$ for $x>0$ on $(-1,1)$, with any value at zero. It is bounded and $C^1$ off zero, with classical derivative zero there. Butso its distributional derivative is a Dirac delta. If a locally integrable weak derivative existed, it would be zero off zero and hence zero almost everywhere, contradicting this identity. The step function is the required counterexample; the cutoff error in dimension one need not vanish.