We know that if $f : X \rightarrow Y $ is a continuous function then image of every compact subset of X is compact in Y. What if we try to prove the converse ? Clearly , image of compact set is compact won't be enough, but can we add other conditions to force the function to be continuous ?
Obviously adding conditions like "f is a homeomorphism" or any such condition which would clearly trivialize the problem won't be allowed.
EDIT X and Y are metric spaces.
I was guessing that imposing the condition "X,Y are infinite and every infinite compact set maps to infinite compact set " might be useful but I can't go any further.