Regular functions on the punctured affine plane

For $W={\mathbb{A}}_k^2\setminus \{0\}$, restriction to $D(x)$ and $D(y)$ givesThe intersection is taken in $k(x,y)$ and the equality follows from unique factorization: a reduced denominator dividing powers of both $x$ and $y$ is a unit. The inclusion $W\hookrightarrow {\mathbb{A}}^2$ induces an isomorphism on global functions but is not an isomorphism, so $W$ is not an affine variety.