TITLE: Covering $\mathbb R^2$ with function graphs
QUESTION [21 upvotes]: Suppose we have a countable family of function graphs (each function is $\mathbb R\to\mathbb R$, not necessary continuous). Obviously, they cannot cover the whole plane $\mathbb R^2$, because they cannot even cover every of uncountably many points on a single vertical line.
But suppose now we are allowed to rotate each graph from the family by arbitrary angle around an arbitrary point of the plane (the total number of graphs is still countable). Is it possible to cover the whole plane $\mathbb R^2$ in this case?

REPLY [7 votes]: You can construct just one function $f: R \rightarrow R$ such that countably many rotations of the graph of $f$ cover the plane. This is due to R. O. Davies. See theorem 31 and the remark below it in Arnie's notes which has many other interesting results (without proofs) about set theory of plane.