MHB Limit of upper semicontinuous function

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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.
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
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