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Math Help - Continuous Function at a Point? Based on Limits

  1. #1
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    Continuous Function at a Point? Based on Limits

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    The Definition of Continuity at a Point is "A function of f is continuous at a point c is f exists at c and (the limit of f(x) as x approaches c from the left) = (the limit of f(x) as x approaches c from the right) = (f(c))"

    My problem with this is I don't know enough to know whether they're trying to trick me by omitting the knowledge if the two limits equal f(1) and, because that's omitted, I should answer with "The point at f(c) does not necessarily exist." or what..

    Could someone explain this to me? Thanks.
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by Savior_Self View Post
    http://i55.tinypic.com/15dmf6a.jpg


    The Definition of Continuity at a Point is "A function of f is continuous at a point c is f exists at c and (the limit of f(x) as x approaches c from the left) = (the limit of f(x) as x approaches c from the right) = (f(c))"

    My problem with this is I don't know enough to know whether they're trying to trick me by omitting the knowledge if the two limits equal f(1) and, because that's omitted, I should answer with "The point at f(c) does not necessarily exist." or what..

    Could someone explain this to me? Thanks.

    You are correct the limits tell you nothing about the values at the point. For example

    f(x)=\begin{cases}x^2, \text{ if } x \ne 0 \\ 29, \text{ if } x=0 \end{cases}

    Both the right and left limit equal zero, but f(0)=29
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  3. #3
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    In other words, the statement is false.
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  4. #4
    Senior Member bkarpuz's Avatar
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    A simple counter example
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