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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?

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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).

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  • $\begingroup$ Thank you @denesp. I was not sure if it was a definition or condition. $\endgroup$
    – eigenvector
    Oct 17, 2017 at 7:39
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    $\begingroup$ @eigenvector That is strange because you yourself wrote "definition" twice. $\endgroup$
    – Giskard
    Oct 17, 2017 at 7:44
  • $\begingroup$ 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? $\endgroup$
    – eigenvector
    Oct 17, 2017 at 7:59
  • $\begingroup$ @eigenvector As I said in my answer it is the definition. $\endgroup$
    – Giskard
    Oct 17, 2017 at 10:05

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