Suppose we have a function from a topological space (X, Tx) to (Y, Ty).
If for each subset of X that is connected (with respect to the induced topology from X) the image of that subset is also connected, does this imply the function must be continuous?
I am pretty sure the reverse of this is true b/c of the Main theorem of connectedness.
If it is true could this than be used as an alternate definition for continuity? It seems like a very intuitive definition to me b/c when I think of a continuous function I think of a mapping that doesn't 'break' apart things that are 'connected'.