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Math Help - finding a maximum and minimum on a compact domain

  1. #1
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    finding a maximum and minimum on a compact domain

    Hello there,
    how would I approach this problem?

    Suppose f be a continuous function on a compact domain D. Show that f has a maximum and a minimum on D.
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by dgmath View Post
    Hello there,
    how would I approach this problem?

    Suppose f be a continuous function on a compact domain D. Show that f has a maximum and a minimum on D.
    compact can be translated as closed and bounded, which means that f has to be a bounded function (you can prove this by contradiction, here is where you use the fact f is continuous). Since f is bounded, the set \{ f(x) \mid x \in D \} has a supremum and infimum. Which are the maximum and minimum values of f respectively.
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