Quote by Bacle2
"now, Is there any constant that is not continuous?"
Try this:
f:X>Y , f(x)=c , constant.
Use the inverse open set definition: V open in Y; then you have two main options:
i)V contains c.
ii)V does not contain c.
What can you say about f^{1}(V)?

The preimage of V has to be either X or the empty set.