In a tutorial I wanted to give a quick explanation of the property of continuity. One of the common intuitions for continuity is that it preserves connection:
Continuous maps do not map connected sets onto disconnected sets, since the restriction of a continuous map to a subspace is again continuous one sees
A continuous map maps every connected subset of a space onto a connected set, in other words continuous maps do not "tear the space apart".
This is nice, but does the reverse also hold? Ie is every map that maps connected subsets to connected sets continuous?
I would think not, but can't think of a counter-example.
Bonus question: Maybe if the domain/image space has certain niceness properties equivalence holds, what could be some examples of such niceness properties?