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Can any one example a function that has limit just in one point

  1. Dec 7, 2008 #1
    can any one example a function that has limit just in one point
     
  2. jcsd
  3. Dec 7, 2008 #2
    Re: limit

    Let f(x)=x if x is rational and f(x)=0 if x is irrational.
     
  4. Dec 7, 2008 #3
    Re: limit

    can you prove it
     
  5. Dec 7, 2008 #4
    Re: limit

    Yup. Pretty easy to see that [tex]\lim_{ x \to 0 } f( x ) = 0[/tex], so we just want to show that it doesn't have a limit anywhere else. To show that it doesn't have a limit at [tex]c\neq0[/tex], take a two sequences approaching c, one along rational numbers and one along irrational numbers, and use the sequential criterion for limits. If you have trouble with that, post what you've tried and I'll fill in the details.
     
  6. Dec 9, 2008 #5
    Re: limit

    You could also use epsilon delta. Only numbers for which e > 0 might not be true are irrational x, but then you can just choose d = e.
     
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