TITLE: Continuity of function mapping connected set to connected set
QUESTION [5 upvotes]: If a function maps every connected set onto a connected set, is it necessarily continuous? I know the converse is true.

REPLY [3 votes]: I assigned a question like this to my students recently. Being continuous in $\mathbb{R}$ is equivalent to the graph being path connected. This is because a path-connected graph is compact on each closed interval, which implies continuity (see Munkres topology).