Let $f: \mathbb R \to \mathbb R$ be any function. When is the graph of $f$ dense in $\mathbb R^2$?  

The only examples I know for this are for non-measurable functions, but is that a necessary condition?



When is the graph of a function a dense set?
The [Conway base 13 function](http://en.wikipedia.org/wiki/Conway_base_13_function) is probably a standard example: the graph is dense because any restriction of the function to an open interval is surjective. But meanwhile, since the graph of the base 13 function is easily defined by an arithmetic property of the digits, the function is Borel and hence measurable. More information is available at [Is Conway's base-13 function measurable?](https://mathoverflow.net/questions/32641/is-conways-base-13-function-measurable/32684#32684)


