Question on regular functions

Hello,

consider the variety N=V(xy-1) in the affine plane A^2 with
algebraically closed ground field. The map f: N --> k defined by (x,y)
|---> 1/x should be a regular function on N because there exists g=1
and h=x in k[x,y] such that
f(x,y)=g(x,y)/h(x,y) for all (x,y) in N and h(x,y) is not zero because
for (x,y)\in V(xy-1) follows that x is not zero.

The identity theorem for affine spaces tells us, that f is the
restriction of a polynomial map F:A^2 -->A^1, but what does the
polynomial F in k[x,y] look like? I mean, what happens with (0,0)?

Thanks
S.
.

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