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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 ?

being_hd
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    No. Such a function does not exist. –  Jul 28 '14 at 15:45
  • could you please format the question in a better way? – user126154 Jul 28 '14 at 15:45
  • What is the nature of the function that I have defined ? – being_hd Jul 28 '14 at 15:47
  • I'm not able to just read your function, please, format it properly – user126154 Jul 28 '14 at 15:48
  • @H.D. I know that you are mainly interested in knowing the answer than in formatting your question. But your question is potentially useful to future visitors of this site. For that reason you should take an effort for making it readable. You can put formulas using standard latex sybols (using dollars) – user126154 Jul 28 '14 at 15:51
  • The function you define in your post seems to have been directly taken from the Wikipedia article. –  Jul 28 '14 at 15:51
  • I've formatted his question correctly; he just needs to accept the edit. – Jam Jul 28 '14 at 15:51
  • @H.D. See also http://math.stackexchange.com/questions/282714/upper-semi-continuous-lower-semi-continuous/323773#323773 for a proof – user126154 Jul 28 '14 at 15:56
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    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. – being_hd Jul 28 '14 at 16:01
  • No Problem HD ! – Nerdy Jul 29 '14 at 05:25

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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 $x=1$.

Jonas Meyer
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