For any f X the following conditions are equivalent 1 f is upper semi continuous 2 f (x) limn f (xn) for every sequence xn x 3 The hypograph of f is closed in X Semi continuity, as opposed to the more restrictive continuity, is often assumed in economic analysis, since it is sufficient to guarantee...

1 implies 2 Suppose is upper semicontinuous Let be a sequence c ...

For any f: X the following conditions are equivalent: 1. f is upper semi continuous.

Question:

For any f: X → ℜ the following conditions are equivalent:
1. f is upper semi continuous.
2. f (x) ≥ limn→∞ f (xn) for every sequence xn → x.
3. The hypograph of f is closed in X × ℜ.
Semi continuity, as opposed to the more restrictive continuity, is often assumed in economic analysis, since it is sufficient to guarantee the existence of a maximum in a constrained optimization model. This is a consequence of the following result, which shows that semi continuous functions obey a form of the Weierstrass theorem.