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Unformatted text preview: , if lim x t = x * then f ( x * ) ≤ lim inf f ( x t ) . Note also that one possible sequence of x t is x t = x * for all t . Informally, a function is lower semicontinuous iﬀ the function only jumps down, if it jumps at all. 1 (c) The epigraph of a function f : R n → R is the set { ( x,y ) ∈ R n +1 : y ≥ f ( x ) } . Informally, the epigraph of f is the set of points lying on or above the graph of f . Prove that f is lower semicontinuous iﬀ the epigraph of f is closed. 1 In case you’ve forgotten the deﬁnition of lim inf: let { x t } be a sequence in R n and let E ⊂ R ∪ {∞ , ∞} be the set of subsequential limits of { x t } . Then lim inf x t = inf E . One can show (the proof is not diﬃcult but I do not require you to provide it) that there is a subsequence { x t k } such that lim x t k = lim inf x t ....
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 Fall '08
 JohnNachbar
 Economics, Topology, Metric space, Compact space, ∈ Rn, lim inf xt

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