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Real Analysis limit problem

  1. Sep 29, 2012 #1
    1. The problem statement, all variables and given/known data
    f:ℝ→ℝ is defined as f(x)= 2x if x is rational and f(x)=4-2x if x is irrational.
    Is it true that lim x→1/2=1?


    2. The attempt at a solution
    Intuitively it seems that as x gets ever closer to 1/2 from either side that the function will oscillate between numbers very close to 1 and 3 and therefore there the limit doesn't exist.
    Firstly is my intuition right?
    Secondly how does one show this using the definition of a limit or some other theorem?
    I thought about using the fact that all (xn)[itex]\subseteq[/itex]ℝ such that (xn)→1/2 would have to imply that (f(xn))→1. Again i can't formally show that this can't be.
     
    Last edited: Sep 29, 2012
  2. jcsd
  3. Sep 29, 2012 #2

    HallsofIvy

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    Yes, that is correct. If x is a rational number close to 1/2, then f(x) is close to 2(1/2)= 1 and if x is an irrational number close to 1/2, then f(x) is close to 4- 2(1/2)= 3. The function does NOT get close to any one number for all real numbers close to 1/2 and so the limit does not exist.
     
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