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Math Help - show f not continous on a

  1. #1
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    show f not continous on a

    For x exsits in real numbers, there is unique n exsists in integers s.t n<=x<n+1. denote this n as [x]. Thus, we obtain a function F: real numbers arrow real numbers, x arrow [x]. e.g. [-2.5]= -3.

    Let a exsist in Integers show f is not continuous at a. ( use suitable sequences which converge to a )??????

    Hope this makes sense thanks.
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  2. #2
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    Quote Originally Posted by ad53ggz View Post
    For x exsits in real numbers, there is unique n exsists in integers s.t n<=x<n+1. denote this n as [x]. Thus, we obtain a function F: real numbers arrow real numbers, x arrow [x]. e.g. [-2.5]= -3.

    Let a exsist in Integers show f is not continuous at a. ( use suitable sequences which converge to a )??????

    Hope this makes sense thanks.
    Condiser the sequnces for a \in \mathbb{Z} x_n=a-\frac{1}{n} and y_n=a+\frac{1}{n} ; n=1,2,3...

    what would f(x_n) equal and f(y_n)

    I hope this helps.

    Good luck.
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  3. #3
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    Quote Originally Posted by TheEmptySet View Post
    Condiser the sequnces for a \in \mathbb{Z} x_n=a-\frac{1}{n} and y_n=a+\frac{1}{n} ; n=1,2,3...

    what would f(x_n) equal and f(y_n)

    I hope this helps.

    Good luck.

    em.. im sorry any chance a bit more in depth here?
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  4. #4
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    surely as n gets infinitely large  x_n=a-\frac{1}{n} tends to  a
     y_n=a+\frac{1}{n} tends to  a


    as a result  f(x_n)=a-1  ???
     f(y_n)=a-1
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  5. #5
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    Quote Originally Posted by ad53ggz View Post
    surely as n gets infinitely large  x_n=a-\frac{1}{n} tends to  a
     y_n=a+\frac{1}{n} tends to  a


    as a result  f(x_n)=a-1 ???
     f(y_n)=a-1
    note that x_n < a for all n and

    y_n > a for all n

    so f(x_n) = a-1 for all n

    f(a)=a

    f(y_n) =a for all n

    so both sequences converge to the same value, but their immages go to two different values.

    Therefore the function is not continous at a when a is an integer.
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  6. #6
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     f(y_n) =a for all n

    thanks for the help!
    can you run this by me, understand the rest and looks like was on the right line but cant see this yet?
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  7. #7
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    Also how would i show f is R-integrable on any interval [a,b] ???
    Last edited by ad53ggz; November 30th 2008 at 11:15 AM.
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