MHB Is Limsup of Upper Semicontinuous Function True? Help Needed

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SUMMARY

The discussion centers on the properties of upper semicontinuous functions and the validity of the statement that the limit superior (limsup) of such functions equals the limit of the function itself. A counterexample provided is the function f(x) = sin(x), which is continuous and has a limsup of 1 as x approaches infinity, demonstrating that the initial claim is false. The conclusion drawn is that limsup f(x) does not necessarily equal lim f(x) for upper semicontinuous functions.

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for any upper semicontinuous function limsup f(x)=lim f(x)...Is thıs true ? I don't know, please help me :)
 
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No,

Just think in $f(x)=sin(x)$, is continuous and its upper limit is 1 when $x\to +\infty$
 

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