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Thread: Examples of bounded/continuous sets and functions

  1. #1
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    Post Examples of bounded/continuous sets and functions

    Give an example of a set X and a function f: X -> Real numbers such that:

    1) f is continuous at 2 points but is discontinuous everywhere else on X.

    I have put this:

    f(x)=
    x^2-x where x belongs to rationals

    0 where x doesn't belong to rationals on X= real numbers


    Is this right?


    2) f is continuous on X and sup {f(x)|x belongs to X} belongs to {f(x)|x belongs to X} but inf{f(x)|x belongs to X} doesn't belong to {f(x)|x belongs to X}

    Could this be a possible answer? :
    f(x) = x^2 on X = (-10,10]


    3) f is bounded on X but not continuous on X

    I'm not sure what to put here.
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  2. #2
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    Re: Examples of bounded/continuous sets and functions

    Quote Originally Posted by CourtneyMoon View Post
    Give an example of a set X and a function f: X -> Real numbers such that:
    1) f is continuous at 2 points but is discontinuous everywhere else on X.
    I have put this:
    f(x)=x^2-x where x belongs to rationals
    0 where x doesn't belong to rationals on X= real numbers
    That will work.
    Quote Originally Posted by CourtneyMoon View Post
    Give an example of a set X and a function f: X -> Real numbers such that:
    2) f is continuous on X and sup {f(x)|x belongs to X} belongs to {f(x)|x belongs to X} but inf{f(x)|x belongs to X} doesn't belong to {f(x)|x belongs to X}
    Could this be a possible answer? :
    $\displaystyle f(x) = x^2$ on $\displaystyle X = (-10,10]$
    The range of your $\displaystyle f(x)$ is $\displaystyle [0,100]$
    Does the range contain both if $\displaystyle \inf~\&~\sup~?$
    Try $\displaystyle f(x)=-x^2$ on $\displaystyle (-1,1).$

    Quote Originally Posted by CourtneyMoon View Post
    Give an example of a set X and a function f: X -> Real numbers such that:
    3) f is bounded on X but not continuous on X
    Let $\displaystyle X=[1,2]$ and $\displaystyle f(x) = \left\{ {\begin{array}{*{20}c} {x,} & {rational} \\ {0,} & {irrational} \\ \end{array} } \right.$.
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  3. #3
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    Post Re: Examples of bounded/continuous sets and functions

    Quote Originally Posted by Plato View Post
    That will work.

    The range of your $\displaystyle f(x)$ is $\displaystyle [0,100]$
    Does the range contain both if $\displaystyle \inf~\&~\sup~?$
    Try $\displaystyle f(x)=-x^2$ on $\displaystyle (-1,1).$


    Let $\displaystyle X=[1,2]$ and $\displaystyle f(x) = \left\{ {\begin{array}{*{20}c} {x,} & {rational} \\ {0,} & {irrational} \\ \end{array} } \right.$.
    Okay thanks, would f(x)= -x^2 on X =(-1,1) (which you said) work on X=(-1,1] too?
    For the f is bounded on X but not continuous on X could I possibly have this:
    f(x)= mod x on X = [-10,10] ?
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  4. #4
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    Re: Examples of bounded/continuous sets and functions

    Quote Originally Posted by CourtneyMoon View Post
    Okay thanks, would f(x)= -x^2 on X =(-1,1) (which you said) work on X=(-1,1] too?
    NO! It does not work for $\displaystyle X=(-1,1]$.
    For now the range is $\displaystyle [-1,0]$ which contains the $\displaystyle \inf$.

    Quote Originally Posted by CourtneyMoon View Post
    For the f is bounded on X but not continuous on X could I possibly have this: f(x)= mod x on X = [-10,10] ?
    What does f(x)= mod x mean?
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  5. #5
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    Re: Examples of bounded/continuous sets and functions

    Hmm okay, thank you.

    Sorry I meant f(x)=|x| on X = [-10,10]
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  6. #6
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    Re: Examples of bounded/continuous sets and functions

    Quote Originally Posted by CourtneyMoon View Post
    Hmm okay, thank you.
    Sorry I meant f(x)=|x| on X = [-10,10]
    NO again! The function $\displaystyle f(x)=|x|$ is continuous everywhere.
    The function I suggested is continuous nowhere on $\displaystyle [1,2]$.
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