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I was asked in a exam: does there exist a function(need not be continous) $f:\mathbb{R}\rightarrow \mathbb{R}$ whose graph is dense in $\mathbb{R}^2$?

I proved that graph of a discontinuous linear map is dense but did not provide explicit example, could any one give me one such? thank you

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For terminating decimals $x=.d_1d_2\dots d_n$, let $f(x)=.d_nd_{n-1}\dots d_2d_1$. For other $x$, define $f$ however you like. This will give a map from $I=[0,1]$ to $I$, dense in $I\times I$. Now compose with any continuous map from $I$ onto the reals, and make $f$ periodic with period 1.

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Very nice example! – Seirios Nov 2 '12 at 9:41
More examples of such functions from $[0,1]$ to $[0,1]$ can be found in the answers to Does there exist a function $f:\[0,1\] \to\[0,1\]$ such its graph is dense in $\[0,1\]\times\[0,1\]$? – Martin Sleziak Nov 2 '12 at 9:42

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