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I don't understand this definition of upper semi-continuity

  1. Oct 16, 2011 #1
    Defn: [itex]f: I → ℝ[/itex] is upper semi-continuous at [itex]x_0 \in I[/itex] if [itex]f(x_0) ≥ \limsup{f(x_0)}[/itex].

    The book goes on to say that "clearly" this is equivalent to saying,

    For any [itex]ε > 0[/itex] there exists a neighborhood [itex]U[/itex] of [itex]x_0[/itex], relative to [itex]I[/itex] such that [itex]f(x) < f(x_0) + ε , \forall x \in U[/itex].

    However, this isn't clear to me. Can someone please explain why these statements are equivalent?
     
    Last edited: Oct 17, 2011
  2. jcsd
  3. Oct 17, 2011 #2

    lurflurf

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    Homework Helper

    It is quite clearly equivalent. What definition of lim sup are you using? If it is not a epsilon-delta definition that may be why it is not obvious. Try writing the lim sup using epsilon-delta.
    Informally we might say
    f is upper semi-continuous if f(x+h) is not more than a little bigger that f(x)
    f is lower semi-continuous if f(x+h) is not more that a little less than f(x)
    f is continuous if f(x+h) is not more than a little different than f(x)

    As the name implies semi-continuous if like half continuous
     
  4. Oct 17, 2011 #3
    Gah. So in the first inequality, limsup f(x0) := limx-->x0 (sup f(x))?

    limsupx-->x0 f(x) = L if for all ε>0 there exists δ>0 such that
    f(x) < L + ε
    whenever |x - x0| < δ

    So, if we take limsup f(x) = L and f is usc at x0
    ==> f(x0) ≥ L

    And by the defn of limsup, for any ε>0,
    ==> L > f(x) - ε

    Then f(x0) ≥ L > f(x) - ε
    ==> f(x0) + ε > f(x)


    Clearly I'm having a lot of trouble with neighborhoods and limsups/liminfs. They haven't quite sunk in yet, I guess.

    Thank you for your help!
     
    Last edited: Oct 17, 2011
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