Topology - Darboux Theorem

Let I be an open interval in and let be a differentiable function.

Let T be the set . I can show that this is a connected subset of with the standard topology.

Let be a function defined by . Prove that .

Show that is an interval.

This last deduction would be trivial if g(T) was a connected set, but I don't see why this should necessarily be true. (It would be true if g was a continuous function but we are not given this. Nevertheless, I think there may be a way of showing it).