Interactive Real Analysis
- part of 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.
Back
Take an arbitrary partition
P = { x0, x1, ..., xn }
of the interval [0, 1].
Back- Between any two points
xj and xj+1
there is an irrational number. Therefore the inf over
[ xj, xj+1 ] must be 0.
That means that L(f, P) = 0.
- Between any two points
xj and xj+1
there is a rational number. Therefore the sup over
[ xj, xj+1 ] must be 1.
That means that
U(f, P) = (x1 - x0) + (x2 - x1) + ... + (xn - xn-1)
which is a telescoping sum so thatU(f, P) = xn - x0 = 1 - 0 = 1