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Maa5616final exampart c autumn 1997 page 2 of 6 2

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MAA5616/FINAL EXAM/PART C Autumn, 1997 Page 2 of 6 2. Define f:[0,1] by f(x) = x if x is irrational or x = 0, and f(x) = x + (1/n) if x 0 is rational and x = m/n in lowest terms. (a) Give an ε - δ proof that f is not continuous at each rational number x ε (0,1]. (b) Give an ε - δ proof that f is continuous at each irrational number x ε (0,1]. (c) Compute the value of the Riemann integral 1 R f(x) dx. 0 Hints: (1) For each positive integer n there are only finitely many rational numbers with denominators less than n in each finite open interval. (2) At some point Proposition 4.7 might be useful. (3) If you want, you may use the Fundamental Theorem of Calculus to evaluate a certain Riemann integral you encounter along the way. Be careful to ensure you have satisfied all the hypotheses, however.
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MAA5616/FINAL EXAM/PART C Autumn, 1997 Page 3 of 6 3. Definition: A real-valued function f is upper semicontinuous at x if, and only if, for each ε > 0, there’s a δ > 0, such that for each t in the domain of f, if t - x < δ , then f(t) < f(x) + ε . Prove Dini’s Theorem: Suppose <f n > is a sequence of upper semicontinuous functions defined on a closed and bounded subset A of with f n (x) 0 as n → ∞ for each x ε A, and f n (x) f n+1 (x)
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