I'm aware that the image of a connected set is connected and the preimage of a disconnected set is disconnected. However, I'm struggling to find an example of a disconnected set such that the image of the disconnected set is also disconnected. Can the preimage of a connected set be disconnected?
2026-04-13 08:57:08.1776070628
Image of a disconnected set is disconnected
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HINT: I assume that you want $f$ to be continuous. Consider the function $f:\Bbb R\to\Bbb R:x\mapsto x^2$. What is the image of $\{-1,1\}$? What is the pre-image of $[1,2]$?