Limit of upper semicontinuous function

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SUMMARY

The discussion centers on the behavior of upper semicontinuous functions concerning limits and limit superior (lim sup). It is established that while the limit may not always exist for upper semicontinuous functions, the lim sup is guaranteed to exist. However, the lim sup does not necessarily equal the function value at the point of upper semicontinuity. For further understanding, the discussion suggests consulting relevant resources, such as Wikipedia.

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  • Understanding of upper semicontinuous functions
  • Familiarity with limit and limit superior concepts
  • Basic knowledge of mathematical analysis
  • Ability to interpret mathematical definitions and theorems
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  • Research the properties of upper semicontinuous functions in detail
  • Study the definitions and applications of limit and lim sup
  • Explore mathematical analysis textbooks for deeper insights
  • Review examples of upper semicontinuous functions and their limits
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Mathematicians, students of mathematical analysis, and anyone studying continuity properties in real analysis will benefit from this discussion.

ozkan12
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for any upper semi continuous how we can take limit ? limit or limsup ? which one is true ?
 
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ozkan12 said:
for any upper semi continuous how we can take limit ? limit or limsup ? which one is true ?

Well, you're definitely not guaranteed to be able to take the limit. The lim sup is guaranteed to exist, but it's not necessarily equal to the function value at which the function is upper semi-continuous. The wiki is fairly helpful, I think.
 

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