Confused about lower semicontinuous functions

Asked Feb 23 '24 at 19:51

Active Feb 23 '24 at 19:51

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LSC is basically : For a function f to be lower semicontinuous at a means that if x is near a then f(x) is greater than or equal to f(a), or at least nearly so.

But in the given graph, f is not LSC at x, as there exists points, call it s, where f(s) is less than f(x)

asked Feb 23 '24 at 19:51

user17420392

1

Unfortunately, your statement following “LSC is basically ...” is wrong. – Martin R Feb 23 '24 at 19:56
I found the statement here and understood LSC based on that – user17420392 Feb 23 '24 at 19:57
Perhaps it depends on what “or at least nearly so” exactly means. – But think of $f(x) = -|x|$ which is continuous (and therefore lower semicontinuous) at $x=0$. – Martin R Feb 23 '24 at 19:59
$f$ is LSC at $a$ if $\liminf_{x \to a} f(x) \ge f(a)$. That does not imply that $f(x) \ge f(a)$ near $a$. – Martin R Feb 23 '24 at 20:01
Partially understood it. If possible, can you tell me how the given function in the image is lsc ? – user17420392 Feb 23 '24 at 20:03
What are the limits at the point in question? – Kurt G. Feb 24 '24 at 05:16

Confused about lower semicontinuous functions

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