What is the relation between continuity and locality if there is any, either in classical or quantum mechanics? Are they the same?
They are different. Continuity is a purely mathematical concept. It says that values do not "jump," but rather change value smoothly.
Locality is different. Locality says that, as time moves forward, the evolution of the state of a point only depends on that which is immediately next to it. It's the principle that says if a magician has sealed a card in a box, there's no way he could write anything on that card because he's not in the box. (And thus, when he does demonstrate that he wrote on it, you assume that either he got into the sealed box, or he took the card out of the box, because there's no way he could have written on it from a distance)