F(x) = x if x is rational, 0 if x is irrational.

  1. 1. The problem statement, all variables and given/known data

    F(x) = x if x is rational, 0 if x is irrational.
    Use the δ, ε definition of the limit to prove that lim(x→0)f(x)=0.
    Use the δ, ε definition of the limit to prove that lim(x→a)f(x) does not exist for any a≠0.


    2. Relevant equations

    lim(x→a)f(x)=L
    0<|x-a|<δ, |f(x)-L|<ε

    3. The attempt at a solution

    I was mostly having troubles writing my initial equation, I was stumped very early on by filling in the values for the epsilon part of the equation, if that's how one is supposed to go by this problem. If not, any other advice that can be given to me?
     
  2. jcsd
  3. SammyS

    SammyS 7,928
    Staff Emeritus
    Science Advisor
    Homework Helper
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    For the limit as x → 0 :
    For rational numbers, f(x) = x.
    What does δ need to be so that if 0 < |x - 0| < δ , then |f(x) - 0| < ε ?

    For irrational numbers, any δ will work. Why?
    So, use the same δ you pick for the rationals.

    Do you know how to show that limx → ag(x) doesn't exist, in general? Of course using δ, ε .
     
  4. I have the exact same exercise and i cannot prove the second part
    "Use the δ, ε definition of the limit to prove that lim(x→a)f(x) does not exist for any a≠0."

    I was wondering if anyone could help me with that

    Thank you in advance
     
  5. D H

    Staff: Mentor

    Assuming a limit did exist, would that lead to any contradictions?
     
  6. Yes, i think that i must assume that the limit exists,lets say that it is L.
    The problem is that i don't know what i have to do next.
    Probably i will have to work with to different cases,one if x is rational and
    one if x is irrational.Another problem is that I don't know what ε Ι have to use
    in order to get to the contradiction.
     
  7. D H

    Staff: Mentor

    Show that no matter what value L you pick as [itex]\lim_{x \to c} f(x)[/itex] means that you can find some [itex]\epsilon>0[/itex] but you cannot find a [itex]\delta>0[/itex] that satisfies the definition of the limit.
     
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