Let $\Omega$ be a bounded, open, simply connected subset of $\mathbb R^n$ with Lipschitz boundary.



> 
> **Question:** Does every function in the Sobolev space $W^{1,1} (\Omega)$
> admit a representative whose graph in $\Omega \times \mathbb R$ has a
> path connected component whose projection to $\Omega$ has full measure in $\Omega$?
> 
> 
> 



Is the graph of a Sobolev function path connected?
As pointed out by Mateusz Kwaśnicki, the answer is yes.


Applying the ACL characterisation of Sobolev functions (see, for example page 36 here: [https://math.aalto.fi/~jkkinnun/files/sobolev\_spaces.pdf](https://math.aalto.fi/%7Ejkkinnun/files/sobolev_spaces.pdf)) to each of the coordinates successively, we obtain a full measure subset $S$ of $\Omega$ whereby any two points in the graph of $f$ over $S$ are joined by the graphs of at most $n$ absolutely continuous functions (that are parallel to the coordinate axes). So in particular the graph over $S$ is path connected, as required.


