= Solution
Let $x_n=(\pi/2+2\pi n)^{-1}$. The points $p_n=(x_n,1)$ lie on the specified graph and converge in $\mathbb R^2$ to $(0,1)$. They form a <Cauchy sequence> in the induced <metric>, but their only possible limit is absent from the space. Adding $(0,0)$ does not supply that missing limit. Hence the specified space is \b[not complete].
For the unheaded request on $\mathbb R$, use the <arctangent pullback metric>
$$
\boxed{d(x,y)=|\arctan x-\arctan y|.}
$$
Positivity, symmetry and the <triangle inequality> follow from the ordinary distance, and injectivity of $\arctan$ gives $d(x,y)=0$ only if $x=y$. The sequence $n$ is Cauchy for this <metric>, because $\arctan n\to\pi/2$, but cannot converge to any finite real $x$: that would require $\arctan x=\pi/2$. Thus this <metric> makes $\mathbb R$ incomplete, even though it induces its usual topology.
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