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Math Help - Cont. Func.

  1. #1
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    Cont. Func.

    Determine whether there exists a function that is continuous on all of \mathbb{R} where it has range f(\mathbb{R}) that's equal to \mathbb{Q}
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  2. #2
    MHF Contributor red_dog's Avatar
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    There isn't such a function, because if f is continuous than f(\mathbf{R}) must be an interval and \mathbf{Q} is not an interval.
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  3. #3
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by red_dog View Post
    There isn't such a function, because if f is continuous than f(\mathbf{R}) must be an interval and \mathbf{Q} is not an interval.
    ? let f(x) = c which continuous on R. is f(\mathbf R) an interval?
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    Quote Originally Posted by kalagota View Post
    ? let f(x) = c which continuous on R. is f(\mathbf R) an interval?
    Continous functions map intervals into intervals or single points. In either case what reddog said is absolutely true.

    EDIT: It would be easier to use the fact that cotinous functions map connected sets to connected sets, R is connected, Q is not.
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