I have a question about the following definition of a continuous preference relation. I apologize for not providing a reference and will try to add one as soon as I can find one.
A preference relation is continuous if for all sequences x_n, y_n with x_n converges to x and y_n converges to y as n goes to infinity and x_n is at least as good as y_n for all n, then this must also hold for the limits x and y.
My simple question is: Is the if an iff or only an if? That is, say this condition (or definition?) above is not ful-filled, can I infer that the preference relation is not continuous or might it still be continuous?