How to prove this expression of lower semi-continuouss?












2












$begingroup$


The definition of lower semi-continuous in wiki (https://en.wikipedia.org/wiki/Semi-continuity) is
begin{equation}
mathop{liminf}_{xto x_{0}} f(x)ge f(x_0).
end{equation}
However, in some books, inequality is changed to equation, i.e.
begin{equation}
mathop{liminf}_{xto 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...






real-analysis convex-analysis







share|cite|improve this question









$endgroup$

















    2












    $begingroup$


    The definition of lower semi-continuous in wiki (https://en.wikipedia.org/wiki/Semi-continuity) is
    begin{equation}
    mathop{liminf}_{xto x_{0}} f(x)ge f(x_0).
    end{equation}
    However, in some books, inequality is changed to equation, i.e.
    begin{equation}
    mathop{liminf}_{xto 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...






    real-analysis convex-analysis







    share|cite|improve this question









    $endgroup$















      2












      2








      2





      $begingroup$


      The definition of lower semi-continuous in wiki (https://en.wikipedia.org/wiki/Semi-continuity) is
      begin{equation}
      mathop{liminf}_{xto x_{0}} f(x)ge f(x_0).
      end{equation}
      However, in some books, inequality is changed to equation, i.e.
      begin{equation}
      mathop{liminf}_{xto 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...






      real-analysis convex-analysis







      share|cite|improve this question









      $endgroup$




      The definition of lower semi-continuous in wiki (https://en.wikipedia.org/wiki/Semi-continuity) is
      begin{equation}
      mathop{liminf}_{xto x_{0}} f(x)ge f(x_0).
      end{equation}
      However, in some books, inequality is changed to equation, i.e.
      begin{equation}
      mathop{liminf}_{xto 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...






      real-analysis convex-analysis




      real-analysis convex-analysis






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 8 '18 at 10:20









      Ze-Nan LiZe-Nan Li

      286




      286






















          1 Answer
          1






          active

          oldest

          votes


















          2












          $begingroup$

          It depends on the definition of $liminf_{xto x_0}$. If you use
          $$
          liminf_{xto x_0} f(x) := lim_{rto 0} inf left{ f(x): d(x,x_0)<r, xne x_0 right}
          $$
          then strict inequality could happen. For example, the function
          $$
          f(x)=cases{0 &; $x=0$\ 1 &; $xinBbb Rbackslash{0}$}
          $$
          is lower semicontinuous at $x=0$ but $liminf_{xto 0} f(x)=1> 0 = f(0)$.



          However, if your definition does not have the restriction $xne x_0$, i.e.
          $$
          liminf_{xto x_0} f(x) := lim_{rto 0} inf left{ f(x): d(x,x_0)<r right}
          $$
          then we can write $liminf_{xto 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.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thank you for your answer. I got it.
            $endgroup$
            – Ze-Nan Li
            Dec 8 '18 at 12:38










          • $begingroup$
            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.
            $endgroup$
            – Dave L. Renfro
            Dec 8 '18 at 17:39






          • 1




            $begingroup$
            Since 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).
            $endgroup$
            – Dave L. Renfro
            Dec 8 '18 at 17:58












          • $begingroup$
            @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.
            $endgroup$
            – BigbearZzz
            Dec 9 '18 at 10:05










          • $begingroup$
            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)
            $endgroup$
            – Dave L. Renfro
            Dec 9 '18 at 10:23













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          1 Answer
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          2












          $begingroup$

          It depends on the definition of $liminf_{xto x_0}$. If you use
          $$
          liminf_{xto x_0} f(x) := lim_{rto 0} inf left{ f(x): d(x,x_0)<r, xne x_0 right}
          $$
          then strict inequality could happen. For example, the function
          $$
          f(x)=cases{0 &; $x=0$\ 1 &; $xinBbb Rbackslash{0}$}
          $$
          is lower semicontinuous at $x=0$ but $liminf_{xto 0} f(x)=1> 0 = f(0)$.



          However, if your definition does not have the restriction $xne x_0$, i.e.
          $$
          liminf_{xto x_0} f(x) := lim_{rto 0} inf left{ f(x): d(x,x_0)<r right}
          $$
          then we can write $liminf_{xto 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.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thank you for your answer. I got it.
            $endgroup$
            – Ze-Nan Li
            Dec 8 '18 at 12:38










          • $begingroup$
            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.
            $endgroup$
            – Dave L. Renfro
            Dec 8 '18 at 17:39






          • 1




            $begingroup$
            Since 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).
            $endgroup$
            – Dave L. Renfro
            Dec 8 '18 at 17:58












          • $begingroup$
            @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.
            $endgroup$
            – BigbearZzz
            Dec 9 '18 at 10:05










          • $begingroup$
            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)
            $endgroup$
            – Dave L. Renfro
            Dec 9 '18 at 10:23


















          2












          $begingroup$

          It depends on the definition of $liminf_{xto x_0}$. If you use
          $$
          liminf_{xto x_0} f(x) := lim_{rto 0} inf left{ f(x): d(x,x_0)<r, xne x_0 right}
          $$
          then strict inequality could happen. For example, the function
          $$
          f(x)=cases{0 &; $x=0$\ 1 &; $xinBbb Rbackslash{0}$}
          $$
          is lower semicontinuous at $x=0$ but $liminf_{xto 0} f(x)=1> 0 = f(0)$.



          However, if your definition does not have the restriction $xne x_0$, i.e.
          $$
          liminf_{xto x_0} f(x) := lim_{rto 0} inf left{ f(x): d(x,x_0)<r right}
          $$
          then we can write $liminf_{xto 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.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thank you for your answer. I got it.
            $endgroup$
            – Ze-Nan Li
            Dec 8 '18 at 12:38










          • $begingroup$
            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.
            $endgroup$
            – Dave L. Renfro
            Dec 8 '18 at 17:39






          • 1




            $begingroup$
            Since 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).
            $endgroup$
            – Dave L. Renfro
            Dec 8 '18 at 17:58












          • $begingroup$
            @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.
            $endgroup$
            – BigbearZzz
            Dec 9 '18 at 10:05










          • $begingroup$
            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)
            $endgroup$
            – Dave L. Renfro
            Dec 9 '18 at 10:23
















          2












          2








          2





          $begingroup$

          It depends on the definition of $liminf_{xto x_0}$. If you use
          $$
          liminf_{xto x_0} f(x) := lim_{rto 0} inf left{ f(x): d(x,x_0)<r, xne x_0 right}
          $$
          then strict inequality could happen. For example, the function
          $$
          f(x)=cases{0 &; $x=0$\ 1 &; $xinBbb Rbackslash{0}$}
          $$
          is lower semicontinuous at $x=0$ but $liminf_{xto 0} f(x)=1> 0 = f(0)$.



          However, if your definition does not have the restriction $xne x_0$, i.e.
          $$
          liminf_{xto x_0} f(x) := lim_{rto 0} inf left{ f(x): d(x,x_0)<r right}
          $$
          then we can write $liminf_{xto 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.






          share|cite|improve this answer









          $endgroup$



          It depends on the definition of $liminf_{xto x_0}$. If you use
          $$
          liminf_{xto x_0} f(x) := lim_{rto 0} inf left{ f(x): d(x,x_0)<r, xne x_0 right}
          $$
          then strict inequality could happen. For example, the function
          $$
          f(x)=cases{0 &; $x=0$\ 1 &; $xinBbb Rbackslash{0}$}
          $$
          is lower semicontinuous at $x=0$ but $liminf_{xto 0} f(x)=1> 0 = f(0)$.



          However, if your definition does not have the restriction $xne x_0$, i.e.
          $$
          liminf_{xto x_0} f(x) := lim_{rto 0} inf left{ f(x): d(x,x_0)<r right}
          $$
          then we can write $liminf_{xto 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.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 8 '18 at 11:40









          BigbearZzzBigbearZzz

          8,72621652




          8,72621652












          • $begingroup$
            Thank you for your answer. I got it.
            $endgroup$
            – Ze-Nan Li
            Dec 8 '18 at 12:38










          • $begingroup$
            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.
            $endgroup$
            – Dave L. Renfro
            Dec 8 '18 at 17:39






          • 1




            $begingroup$
            Since 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).
            $endgroup$
            – Dave L. Renfro
            Dec 8 '18 at 17:58












          • $begingroup$
            @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.
            $endgroup$
            – BigbearZzz
            Dec 9 '18 at 10:05










          • $begingroup$
            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)
            $endgroup$
            – Dave L. Renfro
            Dec 9 '18 at 10:23




















          • $begingroup$
            Thank you for your answer. I got it.
            $endgroup$
            – Ze-Nan Li
            Dec 8 '18 at 12:38










          • $begingroup$
            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.
            $endgroup$
            – Dave L. Renfro
            Dec 8 '18 at 17:39






          • 1




            $begingroup$
            Since 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).
            $endgroup$
            – Dave L. Renfro
            Dec 8 '18 at 17:58












          • $begingroup$
            @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.
            $endgroup$
            – BigbearZzz
            Dec 9 '18 at 10:05










          • $begingroup$
            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)
            $endgroup$
            – Dave L. Renfro
            Dec 9 '18 at 10:23


















          $begingroup$
          Thank you for your answer. I got it.
          $endgroup$
          – Ze-Nan Li
          Dec 8 '18 at 12:38




          $begingroup$
          Thank you for your answer. I got it.
          $endgroup$
          – Ze-Nan Li
          Dec 8 '18 at 12:38












          $begingroup$
          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.
          $endgroup$
          – Dave L. Renfro
          Dec 8 '18 at 17:39




          $begingroup$
          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.
          $endgroup$
          – Dave L. Renfro
          Dec 8 '18 at 17:39




          1




          1




          $begingroup$
          Since 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).
          $endgroup$
          – Dave L. Renfro
          Dec 8 '18 at 17:58






          $begingroup$
          Since 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).
          $endgroup$
          – Dave L. Renfro
          Dec 8 '18 at 17:58














          $begingroup$
          @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.
          $endgroup$
          – BigbearZzz
          Dec 9 '18 at 10:05




          $begingroup$
          @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.
          $endgroup$
          – BigbearZzz
          Dec 9 '18 at 10:05












          $begingroup$
          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)
          $endgroup$
          – Dave L. Renfro
          Dec 9 '18 at 10:23






          $begingroup$
          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)
          $endgroup$
          – Dave L. Renfro
          Dec 9 '18 at 10:23




















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