For x ε a and every n then the convergence is

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0 for x ε A and every n. Then the convergence is uniform on A. Hint: Let ε > 0. Let U n = {x ε A: f n (x) < ε }. Show that for each n, U n = A O n , for some open set O n . You must have U n U n+1 . [Why?] Use ’closed and bounded’ to produce an ’N’. Be very careful with the details.
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MAA5616/FINAL EXAM/PART C Autumn, 1997 Page 4 of 6 4. Suppose that f:[a,b] is continuous and that for some pair of numbers x and y in (a,b) with x < y we have f(x) > f(y). Show that if D + f(x) > 0, then there is a number x 0 in (x,y) with D + f(x 0 ) 0. Hint: Define h(t) = [f(x) - f(t)]/[x - t] for t ε (x,y]. Find the t where h(t) = 0 which is farthest to the right.
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MAA5616/FINAL EXAM/PART C Autumn, 1997 Page 5 of 6 5. One point where beginning measure theory students get confused is distinguishing between a measurable function f which is continuous almost everywhere and a function f with the property that there is a continuous function g such that the set {x: f(x) g(x)} has measure zero. (a) Construct an example of a function f:[0,1] which is continuous nowhere and a function g:[0,1] which is continuous on all of [0,1], such that m({x: f(x) g(x)}) = 0.
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