Prove whether or not the general version of the IVT holds when the range $\displaystyle \mathbb{R}$ is given each of the following topologies:
1. The trivial topology
2. The discrete topology
3. The Lower limit topology
Let $\displaystyle f:[a,b] \rightarrow \mathbb{Re}$ be a continuous function that IVT holds.
Since the image of a (path) connected space under a continuous function is (path) connected, IVT does not hold for (2) and (3) ($\displaystyle \mathbb{R}$ with a discrete and a lower limit topology are totally disconnected).
So, I think IVT holds for 1 only.