One direction is definition of continuity, what about the other direction?
I'm honestly not sure myself, but I would try to show that: at least one partial derivative in at least one coordinate is nonzero → it is localy injective → it is open → it is quotient.
OT: I'm having trouble coming up with surjective continuous map between euclidean spaces that is not open. Is there some theorem about it? It would mean that every such map is quotient, it seems. If the map was not subjective, it would be easy.
Another OT, but vaguely related: If a countinuous map between euclidean spaces was open on any open subspace of domain, would it imply all that map would be open? There doesn't seem to be any way of continuously passing from open map on a subspace to a place where it takes open subset to non-open subset. So the OP would need to just check if some nice subspace is open.