Let f X rightarrow R infinity be a function for some metric

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}?

Solution

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   .