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

  • Thread starter moxy
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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?
 
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  • #2
lurflurf
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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
 
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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!
 
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