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'.
Just to take this off the unanswered list:
Consider real functions $f:\mathbb R\to \mathbb R$. The connected subsets of $\mathbb R$ are the intervals. A function satisfying the intermediate value property takes intervals to intervals, but there are highly discontinuous such functions, as noted in the comments. Such functions are often called Darboux functions.
Looking at the example of $\sin(\frac 1x)$, very close points (i.e lying in a small neighborhood of zero) are kept infinitely far apart in the graph of $x\mapsto \sin(\frac 1x)$.