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If f is bounded and integrable on ...
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Show directly from the definition of uniform continuity (without using Theorem 5 of Appendix III) that a function f uniformly continuous on a closed, finite interval is necessarily bounded there....
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Use the definition of uniform continuity given in the paragraph preceding Theorem 4 to prove that f .x/ D px is uniformly continuous on ...
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If f is bounded and integrable on ...
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Prove that if f and g are bounded and integrable on the interval ...
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Prove that if f is bounded and integrable on an interval containing a, b, and c, then Z b a f .x/ dx C Z c b f .x/ dx D Z c a f .x/ dx...
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Prove that if f and g are bounded and integrable on ...
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Prove that I and I defined in the paragraph following Theorem 2 satisfy I  I as claimed there....
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Let f .x/ D 1=n if x D m=n, where m, n are integers having no common factors, and let f .x/ D 0 if x is an irrational number. Thus, f .1=2/ D 1=2, f .1=3/ D f .2=3/ D 1=3, f .1=4/ D f .3=4/ D 1=4, and so on. Show that f is integrable on ...
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Let f .x/ D n 1 if x D 1=n; n D 1; 2; 3; : : : 0 for all other values of x. Show that f is integrable over ...

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