This painful question is inspired by the question "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.

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.