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Math Help - lower semi continuous function

  1. #1
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    lower semi continuous function

    Hello !

    I want to prove that the supremum of continuous functions is a lower semi continuous function .

    Could someone help me please ?I tried to find on the internet something to help me ,but i had no luck!
    Last edited by asterias; March 19th 2011 at 09:40 AM.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    The result is also valid for the supremum f=\sup \{f_i:\;i\in I\} of lower semicontinuous functions f_i:X\to \mathhbb{R} , (X,T) topological space .

    Hint:

    G_i=f_i^{-1}(\alpha,+\infty)\in T for all i\in I so, G=\bigcup\limit_{i\in I}G_i \in T

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  3. #3
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    Thank you very much for your help :-)
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