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

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- Apr 9th 2009, 02:28 PMAndreametIVT with different topologies
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 - Apr 10th 2009, 10:27 PMaliceinwonderland
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.