The definition of lower semi-continuous in wiki (https://en.wikipedia.org/wiki/Semi-continuity) is \begin{equation} \mathop{\lim\inf}_{x\to x_{0}} f(x)\ge f(x_0). \end{equation} However, in some books, inequality is changed to equation, i.e. \begin{equation} \mathop{\lim\inf}_{x\to x_{0}} f(x) = f(x_0). \end{equation} I don't know the latter one is right or not, and i can't give a counterexample to prove that it is incorrect...
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It depends on the definition of $\liminf_{x\to x_0}$. If you use $$ \liminf_{x\to x_0} f(x) := \lim_{r\to 0} \inf \left\{ f(x): d(x,x_0)<r, x\ne x_0 \right\} $$ then strict inequality could happen. For example, the function $$ f(x)=\cases{0 &; $x=0$\\ 1 &; $x\in\Bbb R\backslash\{0\}$} $$ is lower semicontinuous at $x=0$ but $\liminf_{x\to 0} f(x)=1> 0 = f(0)$.
However, if your definition does not have the restriction $x\ne x_0$, i.e. $$ \liminf_{x\to x_0} f(x) := \lim_{r\to 0} \inf \left\{ f(x): d(x,x_0)<r \right\} $$ then we can write $\liminf_{x\to x_{0}} f(x) = f(x_0) $ because we'd have $\inf \left\{ f(x): d(x,x_0)<r \right\}\le f(x_0)$ which implies that the equality sign actually holds.
BigbearZzz
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Thank you for your answer. I got it. – Zenan Li Dec 08 '18 at 12:38
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For some references where non-deleted neighborhoods are used in defining $\limsup$ and $\liminf$ (your second formulation), see my comment to Question about the definition of lim sup and lim inf on real valued functions. – Dave L. Renfro Dec 08 '18 at 17:39
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1Since my comment from 5 August 2014 didn't specify the books sufficiently for someone not already familiar with them, I'll be more specific here: Theory of Functions of a Real Variable by I. P. Natanson (Volume II, 1960, pp. 149-150) AND Theory of Functions of Real Variables by Henry P. Thielman (1953, pp. 99-100) AND Introduction to Real Functions and Orthogonal Expansions by Béla Sz.-Nagy (1965, pp. 54-58). – Dave L. Renfro Dec 08 '18 at 17:58
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@DaveL.Renfro Thank you, I didn't really know the source for these different formulations. I only recalled that my professor told me about the existence of another one. – BigbearZzz Dec 09 '18 at 10:05
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I might have seen this ambiguity many years ago, but in recent years I became aware of this ambiguity when I was preparing for a talk I that gave in October 2013, and at one point I spent maybe a frantic hour the day before (when I wrote the slides for the talk; before this I had only made some very brief outline notes), trying to resolve an inconsistency that I was coming up with for some basic properties of semi-continuous functions. (continued) – Dave L. Renfro Dec 09 '18 at 10:23
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I finally realized, and yes it took me about an hour (!!) because I thought I was doing something wrong, that the inconsistency had to do with the fact that a few books I had consulted used a different definition of $\liminf$ and $\limsup,$ something that I didn't consider would be the case until after about an hour of panic. – Dave L. Renfro Dec 09 '18 at 10:25
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I can safely say that I have seen more than 4 different notions of upper semicontinuity for multifunctions... – BigbearZzz Dec 09 '18 at 10:29