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Math Help - examples needed

  1. #1
    Member
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    examples needed

    Hi y'all I was just wondering if anyone could help me out with some examples of the following:
    a function f where two of the following conditions hold:
    --f is continuous on [a,b],
    --the interval [a,b] is closed,
    --the interval [a,b] is bounded,
    yet f is not bounded on [a,b].

    a function f where two of the previous conditions hold and f IS bounded on [a,b].

    a function f where \to \mathbb{R}" alt="f\to \mathbb{R}" /> with D\subseteq \mathbb{R} closed and bounded which does not attain its maximum value in D.

    Thanks in advance for any help!
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by dannyboycurtis View Post
    Hi y'all I was just wondering if anyone could help me out with some examples of the following:
    a function f where two of the following conditions hold:
    --f is continuous on [a,b],
    --the interval [a,b] is closed,
    --the interval [a,b] is bounded,
    yet f is not bounded on [a,b].

    a function f where two of the previous conditions hold and f IS bounded on [a,b].

    a function f where \to \mathbb{R}" alt="f\to \mathbb{R}" /> with D\subseteq \mathbb{R} closed and bounded which does not attain its maximum value in D.

    Thanks in advance for any help!
    Well for the first one, it can't be a function in \mathbb{R}, because continuous functions on compact sets are bounded, so we have to be more adventurous.

    Define a set A=\{x\in\mathbb{Q}~|~2\leq x^2\leq3\} (This is basically the interval [\sqrt{2},\sqrt{3}], but since we're in \mathbb{Q}, we have to be more careful when defining it.)

    Then define the function f:A\subset\mathbb{Q}\longrightarrow\mathbb{Q} given by f(x)=\frac{1}{3-x^2}

    EDIT: Just noticed that only TWO of the conditions have to hold. In that case, take f(x)=\frac{1}{x} on (0,1].

    --------

    For the second one, the continuous function a,b)\subset\mathbb{R}\longrightarrow\mathbb{R}" alt="fa,b)\subset\mathbb{R}\longrightarrow\mathbb{R}" /> given by f(x)=x will be bounded on (a,b).

    ----------

    I'm not quite sure I understand what the third question is asking, but what about this:

    =[0,2]\subset\mathbb{R}\longrightarrow\mathbb{R}" alt="f=[0,2]\subset\mathbb{R}\longrightarrow\mathbb{R}" /> given by:

    f(x)=\left\{\begin{array}{lr}x:&0\leq x<1\\0:&1\leq x\leq 2\end{array}\right\}

    f has a \sup in D, but no max.
    Last edited by redsoxfan325; October 30th 2009 at 11:15 PM.
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