Let f: X rightarrow R {infinity} be a function, for some metric space X. We defi
ID: 3401384 • Letter: L
Question
Let f: X rightarrow R {infinity} be a function, for some metric space X. We define "regularizations" f_(-), f_(s) off by f_(-)(x): = sup{g(x): g lessthanorequalto f, g: X rightarrow R {infinity} lower semicontinuous} f_(s)(x): = sup{g(x): g lessthanorequalto f, g: X rightarrow R continuous}. Construct examples where f(-) and f_(s) are not continuous. Does one always have f_(-) = f_(s)? What is the relation between f_(-), f_(s) and f_, defined by f_(x): = lim_y rightarrow x inf f(y): = inf{lim_n rightarrow infinity inf f(y_n): y_n rightarrow x}?Explanation / Answer
b.
b. The ceiling function f(x) of x is a Lower Semi continuous function but not Upper semi continuous . For continuity of any function at a point it is required for it to be Lower semi continuous as well as Upper Semi continuous at that point .
Let fn(x) be defined as 0 , x<= 0
fn(x) = nx , 0<x<1/n
1 , 1/n <= x
Let supremum f(x) = Sup fn(x) then
1 , x<= 0
f(x) =
0 , 0<x
Which is discontinuous .
x belongs to R .
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