104_Fa08_hw_11_sol

104_Fa08_hw_11_sol - Homework 11 for MATH 104 Solution...

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Homework 11 for MATH 104 Solution Problem 1 Recall that the rational ruler function g : (0 , 1) R is deﬁned as g ( x ) = 1 q if x = p q with p Z , q Z \ { 0 } relatively prime , 0 if x is irrational or x = 0. Show (without using the Riemann-Lebesgue Theorem) that g is Riemann integrable. Solution. We apply the Riemann Initegrability Criterion. Let ε > 0. By density of the irrationals, for every partition P of [0 , 1], L ( g , P ) = 0. Therefore, it su ces to show that we can ﬁnd a partition P such that U ( g , P ) < ε . Choose n large enough such that 1 / n < ε/ 2. Consider the partition P = { 0 , 1 / n 2 , 2 / n 2 , . . . , 1 } . We count the intervals in the partition where g takes a value of at least 1 / n . A crude estimate gives one interval where g takes value 1, at most two intervals where g takes value 1 / 2, . . . , at most n intervals where g takes value 1 / n . This yields at most 1 + 2 + ··· + n = n ( n + 1) / 2 such intervals overall. The total contribution of these intervals to

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104_Fa08_hw_11_sol - Homework 11 for MATH 104 Solution...

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