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Homework Help: If f is continuous at c and f(c)>1

  1. Mar 20, 2009 #1
    1. The problem statement, all variables and given/known data

    Hi everyone, this is probably an easy question but I'm having trouble on the wording of the proof.

    Let f be continuous at x=c and f(c) > 1

    Show that there exists an r > 0 such that [tex]\forall x \in B(c,r) \bigcap D : f(x) > 1[/tex]

    2. Relevant equations

    [tex]\forall \epsilon > 0, \exists r > 0, \forall x \in B(c,r) \bigcap D \Rightarrow |f(x) - f(c)| < \epsilon [/tex]

    f(c) > 1

    3. The attempt at a solution

    I'd put something here but I don't really have any attempts. I originally thought of using the limit rule that lim f(x) = f(c) but that doesn't work since c could be an isolated point.

    If I used a graph it just seems so obvious, but I have to use words to prove it. I tried also to use a contradiction, but using just what I put down in (2) didn't lead to any direct contradiction that I could see.

    A nudge in the right direction would be helpful.
  2. jcsd
  3. Mar 20, 2009 #2

    matt grime

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    If in doubt try an example.

    Let's say for the sake of argument that f(c)=2. What choice of epsilon, and r, from 2. above will mean that f(x) > 1 in the ball around c?
  4. Mar 20, 2009 #3


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    f(c)=1+h^2 for some h real, do you understand how to pick e such that f(x)>1.
  5. Mar 20, 2009 #4
    by this reasoning couldn't I just claim that f(c) = 1 + e?
  6. Mar 20, 2009 #5

    matt grime

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    Yes, you can do that.
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