The two-cell hat and quadratic functions belong to H¹₀

Put $\zeta (t)=1-2\sigma (t)$. Formula [F4] gives $\sigma (1-t)=1-\sigma (t)$ on $[0,1]$, hence $\zeta (1-t)=-\zeta (t)$. Let $T(t)=1-t$ on $I$; it is a $C^1$ diffeomorphism with $\mathrm{\mid }{T}^{\mathrm{\prime }}(t)\mathrm{\mid }=1$. The continuous functions ${\zeta }_{+}=\max (\zeta ,0)$ and ${\zeta }_{-}=\max (-\zeta ,0)$ satisfy $0\le {\zeta }_{\pm }\le 1$, and [F10] gives ${\int }_I{\zeta }_{\pm }\le {\int }_I{1}_I={\lambda }_1(I)=1$, so both are integrable. Applying [F6] separately to these nonnegative functions gives ${\int }_I{\zeta }_{+}={\int }_I{\zeta }_{+}(1-t)={\int }_I{\zeta }_{-}$, where the last equality uses antisymmetry. By [F9], ${\int }_I\zeta ={\int }_I{\zeta }_{+}-{\int }_I{\zeta }_{-}=0$. For $0<\epsilon <1\mathrm{/}6$, define $g_{\epsilon }(x)=2\zeta ((x-1\mathrm{/}2+\epsilon )\mathrm{/}(2\epsilon ))$ and $q_{\epsilon }(x)={\int }_0^x{g}_{\epsilon }(t)dt$. By [F4]–[F5], $g_{\epsilon }$ is smooth, equals $2$ for $x\le 1\mathrm{/}2-\epsilon$ and $-2$ for $x\ge 1\mathrm{/}2+\epsilon$. Applying [F6] separately to ${\zeta }_{+}$ and ${\zeta }_{-}$ under $x=1\mathrm{/}2-\epsilon +2\epsilon t$, and subtracting by [F9], gives the transition integral $4\epsilon {\int }_0^1\zeta =0$; the two constant pieces cancel, so ${\int }_0^1{g}_{\epsilon }=0$. For $h\mathrm{\ne }0$ with $x+h\in [0,1]$, the difference quotient of $q_{\epsilon }$ is the average of $g_{\epsilon }$ between $x$ and $x+h$; its difference from $g_{\epsilon }(x)$ is bounded by ${\sup }_{\mathrm{\mid }t-x\mathrm{\mid }\le \mathrm{\mid }h\mathrm{\mid }}\mathrm{\mid }{g}_{\epsilon }(t)-g_{\epsilon }(x)\mathrm{\mid }$, which tends to zero by continuity. Thus $q_{\epsilon }^{\mathrm{\prime }}=g_{\epsilon }$ and $q_{\epsilon }$ is smooth. It vanishes at both endpoints and agrees with $\varphi$ outside $(1\mathrm{/}2-\epsilon ,1\mathrm{/}2+\epsilon )$. On that interval, $\mathrm{\mid }{q}_{\epsilon }^{\mathrm{\prime }}-{\varphi }^{\mathrm{\prime }}\mathrm{\mid }\le 4$ almost everywhere, so $\mathrm{\parallel }{q}_{\epsilon }^{\mathrm{\prime }}-{\varphi }^{\mathrm{\prime }}{\mathrm{\parallel }}_2^2\le 32\epsilon$. Also $\mathrm{\mid }{q}_{\epsilon }-\varphi \mathrm{\mid }\le 8\epsilon$ there and the difference vanishes outside, so $\mathrm{\parallel }{q}_{\epsilon }-\varphi {\mathrm{\parallel }}_2^2\le 128{\epsilon }^3$.

By [F3], each $q_{\epsilon }$ has weak derivative $g_{\epsilon }$. The convergence estimates in step 1.1 and [F8] allow passage to the limit in ${\int }_I{q}_{\epsilon }{\psi }^{\mathrm{\prime }}=-{\int }_I{g}_{\epsilon }\psi$ for every $\psi \in C_c^{\mathrm{\infty }}(I)$, giving ${\int }_I\varphi {\psi }^{\mathrm{\prime }}=-{\int }_I{\varphi }^{\mathrm{\prime }}\psi$, where ${\varphi }^{\mathrm{\prime }}=2$ on the left half and ${\varphi }^{\mathrm{\prime }}=-2$ on the right half. Thus $\varphi \in W^{1,2}(I)$. The polynomial $p$ is smooth with $p^{\mathrm{\prime }}=1\mathrm{/}2-x$ and both $p,p^{\mathrm{\prime }}\in L^2(I)$, so [F1] and [F3] give $p\in W^{1,2}(I)$.

Let $C_{\sigma }={\sup }_{[0,1]}\mathrm{\mid }{\sigma }^{\mathrm{\prime }}\mathrm{\mid }<\mathrm{\infty }$ by [F7], and define ${\eta }_{\epsilon }(x)=\sigma (x\mathrm{/}\epsilon -1)\sigma ((1-x)\mathrm{/}\epsilon -1)$. By [F4]–[F5], this smooth cutoff vanishes for $x\le \epsilon$ and $x\ge 1-\epsilon$, equals $1$ on $[2\epsilon ,1-2\epsilon ]$, and $\mathrm{\mid }{\eta }_{\epsilon }^{\mathrm{\prime }}\mathrm{\mid }\le 2C_{\sigma }\mathrm{/}\epsilon$. For the same $\epsilon$, $q_{\epsilon }=2x$ on $(0,2\epsilon )$ and $q_{\epsilon }=2(1-x)$ on $(1-2\epsilon ,1)$. Thus ${\eta }_{\epsilon }{q}_{\epsilon }\in C_c^{\mathrm{\infty }}(I)$, $\mathrm{\mid }{q}_{\epsilon }\mathrm{\mid }\le 4\epsilon$ and $\mathrm{\mid }{q}_{\epsilon }^{\mathrm{\prime }}\mathrm{\mid }=2$ on $E_{\epsilon }=(0,2\epsilon )\cup (1-2\epsilon ,1)$. The function error has squared norm at most $64{\epsilon }^3$. For the derivative difference $({\eta }_{\epsilon }-1)q_{\epsilon }^{\mathrm{\prime }}+{\eta }_{\epsilon }^{\mathrm{\prime }}{q}_{\epsilon }$, the first term has squared norm at most $16\epsilon$; the second has modulus at most $8C_{\sigma }$ on a set of length $2\epsilon$, so its squared norm is at most $128C_{\sigma }^2\epsilon$. Therefore ${\eta }_{\epsilon }{q}_{\epsilon }-q_{\epsilon }\to 0$ in $W^{1,2}(I)$. Together with step 1.1 this gives ${\eta }_{\epsilon }{q}_{\epsilon }\to \varphi$.

Using the cutoff ${\eta }_{\epsilon }$ from step 2.2, its product with $p$ lies in $C_c^{\mathrm{\infty }}(I)$. Since $\mathrm{\mid }p(x)\mathrm{\mid }\le \min (x,1-x)\mathrm{/}2$ and $\mathrm{\mid }{p}^{\mathrm{\prime }}(x)\mathrm{\mid }\le 1\mathrm{/}2$, $\mathrm{\parallel }{\eta }_{\epsilon }p-p{\mathrm{\parallel }}_2^2\le 4{\epsilon }^3$. The derivative difference is $({\eta }_{\epsilon }-1)p^{\mathrm{\prime }}+{\eta }_{\epsilon }^{\mathrm{\prime }}p$; the first term has squared norm at most $\epsilon$, and the second is supported on two intervals of total length $2\epsilon$ with modulus at most $2C_{\sigma }$, so its squared norm is at most $8C_{\sigma }^2\epsilon$. Hence ${\eta }_{\epsilon }p\to p$ in $W^{1,2}(I)$.

Take the explicit sequence ${\epsilon }_n=1\mathrm{/}(10(n+1))$, which lies in $(0,1\mathrm{/}6)$ and tends to zero. Steps 3.1 and 2.2 approximate $p$ and $\varphi$ by members of $C_c^{\mathrm{\infty }}(I)$ along this sequence. By the closure definition [F2], both classes belong to $W_0^{1,2}(I)$, as claimed.