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Thread: Supremum and maximum

  1. #1
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    Supremum and maximum

    Can anyone provide an example of a differentiable function f(x) such that a supremum of f exists but a maximum of f does not exist, for 0\leq x \leq k (k is any fixed number)?

    Thanks very much.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by WWTL@WHL View Post
    Can anyone provide an example of a differentiable function f(x) such that a supremum of f exists but a maximum of f does not exist, for 0\leq x \leq k (k is any fixed number)?

    Thanks very much.
    Wait, this question doesn't make sense. It is wrong. Are you sure that it's the right question?
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  3. #3
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    It's not a question I've been set, but it's just something I've been thinking about.

    So I guess you're telling me there is no such function? Could you kindly explain why?
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by WWTL@WHL View Post
    It's not a question I've been set, but it's just something I've been thinking about.

    So I guess you're telling me there is no such function? Could you kindly explain why?
    The real reasoning? Continuous(and thus differentiable) functions preserve compactness. Or, more simply Extreme value theorem - Wikipedia, the free encyclopedia
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