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Making sense of continuity at a point where f(x) = Infinity?

  1. Aug 15, 2010 #1
    Is there a way to make sense of the following statement: "[itex]f[/itex] is continuous at a point [itex]x_0[/itex] such that [itex]f(x_0) = \infty[/itex]?" The standard definition of continuity seems to break down here: For any [itex]\epsilon > 0[/itex], there is no way to make [itex]|f(x_0) - f(x)| < \epsilon[/itex], since this is equivalent to making [itex]|\infty - f(x)| < \epsilon[/itex], which cannot happen, since [itex]\infty - y = \infty[/itex] for every [itex]y\in \mathbb R[/itex] and [itex]\infty - \infty[/itex] is undefined. So is there any way to make sense of continuity of an extended real-valued function at a point where it's infinite?
  2. jcsd
  3. Aug 15, 2010 #2


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    Could you try regarding [itex]f(x_0) = \infty[/itex] as [itex]1/f(x_0) = 0[/itex], and so make

    [tex]\left|\frac{1}{f(x_0)}-\frac{1}{f(x)} \right| < \epsilon[/tex]
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