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Showing a function defined on the integers is continuous

  1. Nov 26, 2012 #1
    1. The problem statement, all variables and given/known data

    Suppose that the function f is defined only on the integers. Explain why it is continuous.

    2. Relevant equations

    The ε/δ definition of continuity at a point c:
    for all ε > 0, there exists a δ > 0 such that |f(x) - f(c)| ≤ ε whenever |x - c| ≤ δ

    3. The attempt at a solution

    I understand this completely when we are defining our function on ℝ. I can't intuitively understand this for the integers. I am thinking that if we choose δ = 1, that somehow guarantees that |f(x) - f(c)| will be small enough, but I'm not sure of this because my intuition is stuck in the reals.
     
  2. jcsd
  3. Nov 26, 2012 #2
    You'd be better off with any δ<1, even δ=0.9 or δ=0.99999 will do. Imagine a drawing of the real line with the integers marked. Draw a circle around any integer c with radius δ. Since δ isn't quite 1, this circle wihh not quite reach c+1 on the one side or c-1 on the other. So the only integer x within this circle is x=c. And |f(x) - f(c)| = |f(c) - f(c)| is smaller than any ε>0.
     
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