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Negating Limits and Continuity

  1. Nov 16, 2005 #1
    Let f:I->R and let c in I. I want to negate the statements: "f has limit L at c" and "f is continuous at c". Are these correct?

    f does not have limit L at c if there exists e>0 such that for some sequence {x_n} converging to c, |f(x_n)-L|>e for every n.

    f is not continuous at c if there exists e>0 such that for some sequence {x_n} converging to c, |f(x_n)-f(c)|>e for every n.

    edit: also, what is the negation of "f has a limit at c"?
     
    Last edited: Nov 16, 2005
  2. jcsd
  3. Nov 16, 2005 #2

    HallsofIvy

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    Yes, those will work, since "for all sequences {a_n} converging to c, the sequence {f(x_n)} converges to L", though not the standard definition of limit, is equivalent to it.

    As far as "f does not have a limit at c" (as opposed to "f does not have limit L at c"), you just have to add "for every L":
    "For every number L, there exist an epsilon> 0 and a sequence {x_n} that converges to c, such that |f(x_n)- L|> epsilon."
     
  4. Nov 16, 2005 #3
    Thanks a lot.
     
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