7.1. Riemann Integral

Examples 7.1.6(d):

Suppose f is the Dirichlet function, i.e. the function that is equal to 1 for every rational number and 0 for every irrational number. Find the upper and lower sums over the interval [0, 1] for an arbitrary partition.
Take an arbitrary partition P = { x0, x1, ..., xn } of the interval [0, 1]. Thus, we have shown that for the Dirichlet function and for any partition P we have that L(f, P) = 0 and U(f, P) = 1.
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