A zero distributional gradient is constant on a connected domain

If $\mathrm{\Omega }=\mathrm{\varnothing }$, choose $c=0$. If $n=0$ and $\mathrm{\Omega }\mathrm{\ne }\mathrm{\varnothing }$, then $\mathrm{\Omega }={\mathbb{R}}^0$ is a singleton and its function $u$ is constant. Henceforth assume $n\ge 1$ and $\mathrm{\Omega }\mathrm{\ne }\mathrm{\varnothing }$. Let $B⋐\mathrm{\Omega }$ be any open ball and choose concentric balls $B⋐B^{\mathrm{\prime }}⋐B^{\mathrm{\prime }\mathrm{\prime }}⋐\mathrm{\Omega }$. By [F1] take $\chi \in C_c^{\mathrm{\infty }}(B^{\mathrm{\prime }\mathrm{\prime }})$ equal to one on $B^{\mathrm{\prime }}$, and extend $w:=\chi u$ by zero to ${\mathbb{R}}^n$. For sufficiently small $\epsilon$, every mollifier centered at a point of $B$ is supported in $B^{\mathrm{\prime }}$. For such a point $x$, differentiating the convolution gives ${\mathrm{\partial }}_i(w\ast {\rho }_{\epsilon })(x)=-{\int }_{B^{\mathrm{\prime }}}u(y){\mathrm{\partial }}_{y_i}{\rho }_{\epsilon }(x-y)dy=0$ by the distributional identity [F2]. Thus $w\ast {\rho }_{\epsilon }$ is constant on the convex ball $B$ by the one-dimensional fundamental theorem along each segment.

Since $w=u$ on $B^{\mathrm{\prime }}$, the approximate-identity property [F4] gives $w\ast {\rho }_{{\epsilon }_j}\to u$ in $L^1(B)$ for a sequence ${\epsilon }_j\downarrow 0$. Write each constant from step 1.1 as $c_j$. The $L^1(B)$ convergence makes $(c_j)$ Cauchy because $\mathrm{\mid }B\mathrm{\mid }\mathrm{\mid }{c}_j-c_k\mathrm{\mid }\le \mathrm{\parallel }{c}_j-u{\mathrm{\parallel }}_{L^1(B)}+\mathrm{\parallel }{c}_k-u{\mathrm{\parallel }}_{L^1(B)}$. Its limit $c_B$ satisfies $u=c_B$ almost everywhere on $B$.

If $\mathrm{\Omega }=\mathrm{\varnothing }$, any constant satisfies the conclusion. Otherwise fix one ball and let $c$ be its almost-everywhere value. Let $S$ be the set of points of $\mathrm{\Omega }$ having a neighborhood on which $u=c$ almost everywhere. It is nonempty and open. If $x\in \bar{S}\cap \mathrm{\Omega }$, choose a ball $B⋐\mathrm{\Omega }$ containing $x$ and a point of $S$; step 2.1 makes $u$ almost everywhere constant on $B$, and its positive-measure intersection with a neighborhood from $S$ forces that constant to be $c$. Thus $x\in S$, so $S$ is relatively closed. Connectedness [F5] gives $S=\mathrm{\Omega }$. By the countable rational base [F6], choose a countable cover of $\mathrm{\Omega }$ by rational balls contained in these neighborhoods; the union of their exceptional null sets is null, hence $u=c$ almost everywhere on all of $\mathrm{\Omega }$. The constants $c_j$ and $c_B$ may be complex; completeness of $\mathbb{C}$ justifies their limits. When $u$ is real almost everywhere on a nonempty domain, its a.e. constant $c$ is real because each ball has positive measure.