Prove whether or not the general version of the IVT holds when the range is given each of the following topologies:
1. The trivial topology
2. The discrete topology
3. The Lower limit topology
Since the image of a (path) connected space under a continuous function is (path) connected, IVT does not hold for (2) and (3) ( with a discrete and a lower limit topology are totally disconnected).
So, I think IVT holds for 1 only.