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.
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Take an arbitrary partition P = { x₀, x₁, ..., x_n } of the interval [0, 1].

Between any two points x_j and x_j+1 there is an irrational number. Therefore the inf over [ x_j, x_j+1 ] must be 0. That means that L(f, P) = 0.
Between any two points x_j and x_j+1 there is a rational number. Therefore the sup over [ x_j, x_j+1 ] must be 1. That means that
U(f, P) = (x₁ - x₀) + (x₂ - x₁) + ... + (x_n - x_n-1)
which is a telescoping sum so that
U(f, P) = x_n - x₀ = 1 - 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.

Interactive Real Analysis, ver. 1.9.5
(c) 1994-2007, Bert G. Wachsmuth
Page last modified: Mar 28, 2007