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Math Help - 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? :
    f(x) = x^2 on X = (-10,10]
    The range of your f(x) is [0,100]
    Does the range contain both if \inf~\&~\sup~?
    Try f(x)=-x^2 on (-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 X=[1,2] and 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 f(x) is [0,100]
    Does the range contain both if \inf~\&~\sup~?
    Try f(x)=-x^2 on (-1,1).


    Let X=[1,2] and 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 X=(-1,1].
    For now the range is [-1,0] which contains the \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 f(x)=|x| is continuous everywhere.
    The function I suggested is continuous nowhere on [1,2].
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