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which of the followings are true?

  1. $X$ be a topological space, $f_n:X\rightarrow \mathbb{R}$ is sequence of lower semi continuous functions then the $\sup\{f_n\}=f$ is also lower semi continuous.
  2. every continuous real valued function on $X$ is lower semi continuous.
  3. A real valued function on $X$ is continuous iff it is both USC and LSC.

I read in my measure theory course and recall that $3$ and $1$ is true though I can not remember the proofs now, but could any one just give me hint how to handle $2$? Thank you.

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Write down the definition of a continuous function and a lower semicontinuous function. Does the first imply the second? –  Ayman Hourieh Jan 20 '13 at 13:09
    
If you believe that 3 is true, what does that tell you about 2? –  mrf Jan 20 '13 at 13:31
    
@mrf $2$ will be false then. –  El Angel Exterminador Jan 20 '13 at 14:04
    
I usually use item 3 as the definition of a continuous function. Intuitively, it is a function that jumps neither up (lower semicontinuity) nor down (upper semicontinuity). Only item 1 needs to be shown with a pencil at hand using definitions. People who study measure theory produce such simple proofs easily, without using any recollections. –  user65491 Mar 7 '13 at 10:41

1 Answer 1

up vote 4 down vote accepted

One definition that can be used for $\small\begin{array}{c}\text{upper}\\\text{lower}\end{array}$-semicontinuity is that $f$ is $\small\begin{array}{c}\text{upper}\\\text{lower}\end{array}$-semicontinuous if and only if $$ \{x:f(x)\lessgtr\alpha\} $$ is open for all $\alpha$.

Hints

  1. Note that $$ \{x:\sup_{n\ge1}f_n(x)\gt\alpha\}=\bigcup_{n=1}^\infty \{x:f_n(x)\gt\alpha\} $$

  2. One definition that can be used for continuity is that $f$ is continuous if and only if $f^{-1}(U)$ is open for all open $U$. Then note that $\{x:f(x)\gt\alpha\}=f^{-1}\left(\{x:x\gt\alpha\}\right)$.

  3. In a fashion similar to 2. we can show that every continuous function is upper-semicontinuous. Thus, we just need to show that each function that is both upper and lower semicontinuous is continuous. Suppose that $f$ is both upper and lower semicontinuous. Then $$ f^{-1}(\alpha,\beta)=\{x:f(x)\gt\alpha\}\cap\{x:f(x)\lt\beta\} $$ is open for all $(\alpha,\beta)$. Furthermore, for every open set, $U$, $$ U=\bigcup_{u\in U}(u-\epsilon_u,u+\epsilon_u) $$ where $\epsilon_u\gt0$ is chosen so that $(u-\epsilon_u,u+\epsilon_u)\subset U$.

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