(The inequality is meant to hold for every sequence x_n converging to x.)
Please note that I did not mean >= or liminf, i.e. upper or lower semicontinuous.
I apologize for answering a question with a question, but why do you suspect this type of function should have a name?
Upper/Lower semicontinuous functions are useful for optimization problems, and producing solutions to elliptic equations. Essentially, if f(x)
I realized that, in a lemma I'm using, it sufficed to assume this condition instead of lower semicontinuity. However, upon reading your reply while watching a Star Wars movie I've noticed both are equivalent.
For if that condition holds for every sequence, it must hold for the subsequences of a given x_n. The images by f of some subsequence converge to liminf f(x_n). Then the limsup for that subsequence is the liminf of the original sequence.
It's odd how the mind works. I don't even like Star Wars.
I truly appreciate your feedback, Justin. I agree that the functions, as described, do not look useful. It was just a matter of generality.
I'm sorry for the confusion about your name. I replied from a smartphone and a very small section of my reply already filled all the screen. Sorry again :(
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