Question: Let f [0, ) g,h [0, ) : R be a bounded function. Consider the functions R that are defined by g(x) = sup

Let f [0, ) g,h [0, ) : R be a bounded function. Consider the functions R that are defined by g(x) = sup {f(y) y x} and h(x) inf {f(y) ya} for every x 0. = (a) (10 points) Explain why g, h are well defined. (b) (20 points) Prove that lim f (x) exists if and only if the limits 818 lim g(x) and lim h (a) exists and lim g(x) = lim h (a). 818 818 888 818

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