# Continuity of preference relation (iff?)

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.

Definition:

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?

## 1 Answer

This is a definition. Definitions are always iff.

Consider this property. From now on we call this and exactly this (name of the concept here).

• Thank you @denesp. I was not sure if it was a definition or condition. – eigenvector Oct 17 '17 at 7:39
• @eigenvector That is strange because you yourself wrote "definition" twice. – Giskard Oct 17 '17 at 7:44
• Fair point. Note, though, that I also wrote: "condition (or definition?) once in the last paragraph. Anyway, lets agree that my wording was not very precise. :-) My understanding is still confused. Let me rephrase my question: Is what I wrote above a definition or is it a condition? – eigenvector Oct 17 '17 at 7:59
• @eigenvector As I said in my answer it is the definition. – Giskard Oct 17 '17 at 10:05