1. The problem statement, all variables and given/known data All right, so this appeared on my final. The intervals are in the reals: If f : [a, b] -> [c, d] , and the graph of f is closed, is f continuous? 2. Relevant equations 3. The attempt at a solution Well, my gut reaction is no, just because it seems like a fairly strong claim, but I couldn't really come up with a decent counterexample. On the other hand, I can see that the graph has to be compact. And I think that would imply that f([a, b]) has to be compact (if it wasn't you could just take any open cover of [a, b] and an open cover of f([a, b]) that doesn't have a finite subcover, then the Cartesian product would be an open cover without finite subcover, I think). But I'm not sure where I can go from there or if that's even useful information.