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A Continuity Problem

  1. Jul 23, 2008 #1
    The problem statement, all variables and given/known data
    Let f be a function with the property that every point of discontinuity is a removable discontinuity. This means that [tex]\lim_{y\to x} f(y)[/tex] exists for all x, but f may be discontinuous at some (even infinitely many) numbers x. Define [tex]g(x) = \lim_{y\to x} f(y)[/tex]. Prove that g is continuous.

    The attempt at a solution
    So I have to prove that for all a,

    [tex]\lim_{x \to a} g(x) = \lim_{x \to a} \lim_{y\to x} f(y) = g(a) = \lim_{y\to a} f(y) [/tex]

    In other words, for every e > 0, there is a d > 0 such that

    [tex]\left| \lim_{y\to x} f(y) - \lim_{y\to a} f(y) \right| < e[/tex]

    for all x satisfying |x - a| < d. I have no clue how to find d. Any tips.
     
  2. jcsd
  3. Jul 23, 2008 #2

    HallsofIvy

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    I think you are going to need to reduce the two limits in your last inequality to their definitions.
     
  4. Jul 23, 2008 #3
    OK. So let d' be such that |f(y) - g(x)| < e for all y with |y - x| < d' and let d'' be such that |f(y) - g(a)| < e for all y with |y - a| < d''.

    The problem now is that I can't fiddle around with |f(y) - g(x)| < e and |f(y) - g(a)| < e because there are different intervals where these inequalities are true.
     
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