= Solution
{c}
Besides $A,B,C$, set $D=(1,2)$, $E=(2,1)$ and $F=(2,2)$. The four triangles are $ABD$, $BCE$, $BDE$ and $DEF$, each of area $1/2$. The nodal function is one at $B$, zero at every other node, and affine on each triangle. The vertex conditions give
$$
\boxed{\phi_B(x,y)=\begin{cases}
2-y,&(x,y)\in ABD,\\
2-x,&(x,y)\in BCE,\\
3-x-y,&(x,y)\in BDE,\\
0,&(x,y)\in DEF.
\end{cases}}
$$
The formulas agree on every shared edge, proving continuity.
\Image[/past-exam-of-the-approximation-theory-and-finite-element-methods-course-of-the-university-of-oxford/2007-four-triangle-mesh.png]
{title=The four triangles and the single free nodal degree of freedom}
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