This painful question is inspired by the question 
"[non-Lie subgroups](https://mathoverflow.net/questions/3157/non-lie-subgroups)" . Let $f$ be an discontinuous additive map from $\mathbb{R} \to \mathbb{R}$. Is it possible that the graph of $f$, inside $\mathbb{R}^2$ with the usual topology, is connected? Remember that the graph of discontinuous function can be connected, as in the 
[Toplogist's sine curve](http://en.wikipedia.org/wiki/Topologist%27s_sine_curve).

I was hoping to show that any connected subgroup of a Lie group is a Lie group, thus answering the previous question, and I couldn't get rid of this horrible example.