Answer Happy

(5) A function f: E → R is lower semicontinuous on E if for every x € E and every ɛ > 0, there exists > 0 such that for all t € E with |t − x| < 8, we have f(x) < f(t) + ɛ. Note: for continuity, one contends with the inequality |f(x) − f(t)| < & which can be rearranged as f(t) - ɛ < f(x) < ƒ(t) +ɛ. Living up to its name, the notion of lower semicontinuity is based on one of the two inequalities. There is a similar definition of upper semicontinuous functions. Consider a sequence (fn)n>0 of lower semicontinuous functions on E. Suppose that for all x € E, the value of supn>o fn(x) is a real number, and let f(x) = sup fn(x) n>0 Prove that f is lower semicontinuous on E.