Continuous matching piecewise polynomials belong to W1p

If $\mathcal{T}=\mathrm{\varnothing }$, then $\mathrm{\Omega }=\mathrm{\varnothing }$ by the mesh definition; $v$ is the unique zero class in $W^{1,p}(\mathrm{\Omega })$ and the cellwise-gradient assertion has no cells. Assume now $\mathcal{T}\mathrm{\ne }\mathrm{\varnothing }$. The mesh is finite and each cellwise polynomial and its ordinary first derivatives are bounded on the closed cell. Thus $v$ and the piecewise function $g_i={\mathrm{\partial }}_i(v{\mathrm{\mid }}_K)$ belong to $L^p(\mathrm{\Omega })$ for every $1\le p\le \mathrm{\infty }$. Fix a coordinate $i$ and a test function $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$. On each cell apply [F2] to $G=(v\phi )e_i$ to obtain ${\int }_{K}v{\mathrm{\partial }}_i\phi dx=-{\int }_K{g}_i\phi dx+{\sum }_{F\subset \mathrm{\partial }K}{\int }_{F}v\phi (n_{K,F}{)}_{i}dS\mathrm{.}$

Sum these finitely many identities over $K$. On an interior facet, [F1] gives a common trace for $v\phi$ and [F3] gives opposite outward normals, so the two flux integrals cancel. On a boundary facet, $\phi =0$ because its support is compact in $\mathrm{\Omega }$. Therefore ${\int }_{\mathrm{\Omega }}v{\mathrm{\partial }}_i\phi dx=-{\int }_{\mathrm{\Omega }}{g}_i\phi dx\mathrm{.}$ By [F4], $g_i$ is the weak $i$th derivative. This holds for every coordinate $i$; all $g_i$ are in $L^p$, so $v\in W^{1,p}(\mathrm{\Omega })$ with the stated cellwise gradient.