3
$\begingroup$

I am looking for an example of a function which is both upper and lower semi continuous but is not continuous. I have an example: $$f(x):=\begin{cases} 1 & \mathrm{if}\; x < 1,\\[7pt] 2 & \mathrm{if}\; x = 1,\\[7pt] \frac{1}{2} & \mathrm{if}\; x > 1. \end{cases}$$ Am I correct ?

$\endgroup$
10
  • 4
    $\begingroup$ No. Such a function does not exist. $\endgroup$ – user61527 Jul 28 '14 at 15:45
  • $\begingroup$ could you please format the question in a better way? $\endgroup$ – user126154 Jul 28 '14 at 15:45
  • $\begingroup$ What is the nature of the function that I have defined ? $\endgroup$ – creative Jul 28 '14 at 15:47
  • $\begingroup$ I'm not able to just read your function, please, format it properly $\endgroup$ – user126154 Jul 28 '14 at 15:48
  • 1
    $\begingroup$ Sorry brothers/sisters, I am just trying to understand these concepts. I have given a try and only then I have asked, just to know whether I am right or wrong. Thank You Dear brothers/sisters for helping me out. I dont know how to use latex, for that reason I could not format the question properly. I will learn it soon. Thanx. $\endgroup$ – creative Jul 28 '14 at 16:01
1
$\begingroup$

A function is continuous if and only if it is upper and lower semicontinuous.

The function you defined is upper semicontinuous but not lower semicontinuous at $x=1$.

$\endgroup$
1

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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