Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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. – La Belle Noiseuse 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
up vote 7 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 $\alpha$.


  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 $(u-\epsilon_u,u+\epsilon_u)\subset U$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.