Consider the Dirichlet function, introduced in Example 5.1.6(g), defined by f(x) := 1 for x ? [0, 1] rational and f(x) := 0 for x ? [0, 1] irrational. Use the preceding exercise to show that f is not Riemann integrable on [0, 1].
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A function f is Riemann integrable on an interval [a, b] if there exists a number I such that for every ε > 0, there exists a partition P of [a, b] such that for every refinement P' of P and for every choice of sample points x_i* in [x_{i-1}, x_i] (i = 1, 2, ..., Show more…
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